ZhuJiaHao
/
zdqy_firm_former


			
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							<!DOCTYPE html>
<html>
<head>
<title>Dynamic Preview of Textarea with MathJax Content</title>
<!-- Copyright (c) 2012-2018 The MathJax Consortium -->
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<meta http-equiv="X-UA-Compatible" content="IE=edge" />

<style>
.changed { color: red }
</style>

<script type="text/x-mathjax-config">
  MathJax.Hub.Config({
    TeX: {
      equationNumbers: {autoNumber: "AMS"},
      extensions: ["begingroup.js"],
      noErrors: {disabled: true}
    },
    showProcessingMessages: false,
    tex2jax: { inlineMath: [['$','$'],['\\(','\\)']] }
  });
//MathJax.Hub.signal.Interest(function (message) {console.log(message,MathJax.Hub.cancelTypeset)});
</script>
<script type="text/javascript" src="../MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>

<script>
var Preview = {
  typeset: null,     // the typeset preview area (filled in by Init below)
  preview: null,     // the untypeset preview    (filled in by Init below)
  buffer: null,      // the new preview to be typeset (filled in by Init below)
  par: [],           // paragraph-specific data
  refs: [],          // undefined references needing to be reprocessed
  updateNeeded: 0,   // number of paragraphs needing update

  oldtext: '',       // used to see if an update is needed
  pending: false,    // true when a restart is in the MathJax queue

  highlight: true,   // color setions that are changed
  noflicker: false,  // reduce flicker
  flickertime: 1500, // processUpdateTime value used for noflicker
  classDelay: 400,   // how long to leave changed paragraphs colored
  ctimeout: null,    // timeout for changed style remover
  labelDelay: 1250,  // how long to wait before reprocessing for label changes
  ltimeout: null,    // timeout for changed labels

  keytimes: [],      // tracks the times between keypresses
  keyrate: 100,      // the average of the keytimes (default value)
  keyn: 0,           // key index to replace next
  keysize:  10,      // use this many keypresses

  //
  //  Get the preview and buffer DIV's
  //
  Init: function () {
    this.typeset = document.getElementById("MathPreview");
    this.buffer = document.createElement("div");
    this.preview = document.createElement("div");
    for (var i = 0; i < this.keysize; i++) {this.keytimes[i] = this.keyrate}
  },

  //
  //  This gets called when a key is pressed in the textarea.
  //
  Update: function (up) {
    if (up) {
      //
      //  Determine the typing speed as a rolling average of the last few keystrokes
      //
      var time = new Date().getTime();
      if (this.lasttime) {
        var delta = time - this.lasttime;
        if (delta < 4*this.keyrate) {
          this.keyrate = (this.keysize*this.keyrate+delta-this.keytimes[this.keyn])/this.keysize;
          this.keytimes[this.keyn++] = delta;
          if (this.keyn === this.keysize) {this.keyn = 0}
        }
      }
      this.lasttime = time;
    }
    //
    //  Get the new text and see if it has changed
    //
    var text = document.getElementById("MathInput").value;
    text = text.replace(/^\s+/,'').replace(/\s+$/,'').replace(/\r\n?/g,"\n");
    if (text !== this.oldtext) {
if (this.rtimeout) {clearTimeout(this.rtimeout)}
      //
      //  Save the text and if we don't already have a pending Restart
      //  clear the timeouts for updating the colors and other updates
      //  then set the update time based on the key rate (so it can be
      //  interrupted better), cancel the current typesetting, if any,
      //  and start a new typesetting run after a short delay (to allow
      //  the editor to process more keystrokes).
      //
      this.oldtext = text;
      if (!this.pending) {
        this.pending = true;
        if (this.ctimeout) {clearTimeout(this.ctimeout); this.ctimeout = null}
        if (this.ltimeout) {clearTimeout(this.ltimeout); this.ltimeout = null}
        MathJax.Hub.processUpdateTime = (this.noflicker ? this.flickertime : this.keyrate / 2);
        MathJax.Hub.Cancel(); this.cancelled = true;
//        MathJax.Hub.Queue(
//          // allow a little time for additional typing
////          ["Delay",MathJax.Callback,Math.min(100,Math.floor(this.keyrate/2)+1)],
//          ["Delay",MathJax.Callback,150],
//          ["Restart",this]
//        );
      }
this.rtimeout = setTimeout(function () {
  this.rtimeout = null;
  MathJax.Hub.Queue(["Restart",Preview]);
},Math.floor(this.keyrate*1.25))
    }
  },

  //
  //  To restart typesetting, touch up the text string
  //  and save it in the buffer.  Then check if an update is
  //  needed, and if so, queue the update.
  //
  Restart: function () {
    this.pending = false;
    var text = this.oldtext.replace(/&/g,'&amp;').replace(/</g,'&lt;').replace(/>/g,'&gt;');
    var text = text.replace(/\n\n+/g,"<p>");
    this.buffer.innerHTML = text;
    var update = this.CompareBuffers();
    if (update.needed) {
      MathJax.Hub.Queue(
        ["CopyChanges",this,update],
        ["PreTypeset",this,update],
        ["Typeset",this,update],
function () {if (MathJax.Hub.cancelTypeset) {console.log("cancel")}},
        ["PostTypeset",this,update]
      );
    }
  },

  CompareBuffers: function () {
    var b1 = this.buffer.childNodes,
        b2 = this.preview.childNodes,
        i, m1 = b1.length, m2 = b2.length, m = Math.min(m1,m2);
    //
    //  Make sure all top-level elements are containers
    //
    for (i = 0; i < m1; i++) {
      var node = b1[i];
      if (typeof(node.innerHTML) === "undefined") {
        this.buffer.replaceChild(document.createElement("span"),node);
        b1[i].appendChild(node);
      }
    }
    //
    //  Find first non-matching element, if any,
    //    and the last non-matching element
    //
    for (i = 0; i < m; i++) {if (b1[i].innerHTML !== b2[i].innerHTML) break}
    if (i === m && m1 === m2) {return {needed: false}}
    while (m1 > i && m2 > i) {if (b1[--m1].innerHTML !== b2[--m2].innerHTML) break}
    return {needed:true, start:i, end1:m1, end2:m2};
  },

  CopyChanges: function (update) {
    var i = update.start, m1 = update.end1, m2 = update.end2, j, par;
    var b1 = this.buffer.childNodes,
        b2 = this.typeset.childNodes;
    update.indices = []; update.nodes = []; update.replace = true;
    //
    //  Remove differing elements from typeset copy
    //  Copy any predeeding old replacements from an interrupted typeset
    //  Add in the new (untypeset) elements
    //  Copy any following old replacements from an interrupted typeset
    //
    this.recordOldData(this.par.splice(i,m2+1-i),i); var tail = b2[m2+1];
    if (this.replace && i > this.start) {
      j = update.start = this.start;
      this.copyPrevious(j,i,i,update);
    }
    while (m2 >= i && b2[i]) {this.typeset.removeChild(b2[i]); m2--}
    while (i <= m1 && b1[i]) {
      this.par.splice(i,0,{number:0, labels:[], defs:[], refs:[], replaced:true, update:true});
      var node = b1[i].cloneNode(true); update.nodes.push(node);
      update.indices.push(i++); this.updateNeeded++;
      if (tail) {this.typeset.insertBefore(node,tail)} else {this.typeset.appendChild(node)}
      if (this.highlight) {this.addChanged(node)}
    }
    if (this.replace && i <= this.end) {
      update.end = this.end;
      this.copyPrevious(i,this.end,update.start,update);
    } else {update.end = i-1};
    //
    //  Swap buffers and set up the new buffer for the next change
    //
    this.preview = this.buffer; this.buffer = document.createElement("div");
  },

  //
  //  Copy data for replaced paragraphs from an interrupted
  //  typesetting run.  Replace the paragraph by their originals
  //  to avoid any problems with partially completed math.
  //
  copyPrevious: function (i,end,start,update) {
    while (i <= end) {
      par = this.par[i];
      if (par.replaced && par.update) {
        this.typeset.replaceChild(this.preview.childNodes[i].cloneNode(true),
                                  this.typeset.childNodes[i]);
        update.nodes.push(this.typeset.childNodes[i]); 
        update.indices.push(i);
      }
      this.recordOldData([par],i,start); par.replaced = true; i++;
    }
  },

  //
  //  Save the labels, defintions, and equation numbers
  //  for the removed paragraphs
  //
  recordOldData: function (par,p,q) {
    var AMS = MathJax.Extension["TeX/AMSmath"];
    var labels = [], defs = [], number = 0;
    for (var i = 0, m = par.length; i < m; i++) {
      if (!par.replace) {
        number += par[i].number;
        defs.push(par[i].defs.all);
        for (var j = 0, n = par[i].labels.length; j < n; j++) {
          delete AMS.labels[par[i].labels[j].split(/=/)[0]];
          labels.push(par[i].labels[j]);
        }
      }
    }
    labels = labels.join(''); defs = defs.join('');
    if (this.replace) {
      this.oldnumber += number;
      if (p < q) {
        this.oldlabels = labels + this.oldlabels;
        this.olddefs   = defs   + this.olddefs;
      } else {this.oldlabels += labels; this.olddefs += defs}
    } else {this.oldlabels = labels; this.olddefs = defs; this.oldnumber = number}
  },

  //
  //  Save the data needed for the incremental typesetting
  //  and reset the TeX begingroup stack
  //
  PreTypeset: function (update) {
    var TEX = MathJax.InputJax.TeX;
    //
    //  Set the state variables
    //
    this.incremental = true; this.cancelled = false;
    this.i = this.j = 0; this.eqNum = 0;
    this.update = update.indices; this.replace = update.replace;
    if (this.replace) {this.start = update.start; this.end = update.end}
    //
    //  Pop any left over \begingroups and push a new one
    //  Reset the equation numbers (but not labels)
    //
    while (TEX.rootStack.top > 1) {TEX.rootStack.stack.pop(); TEX.rootStack.top--}
    TEX.rootStack.Push(TEX.nsStack.nsFrame());
    MathJax.Hub.cancelTypeset = false;   // won't need to do this in the future
  },

  //
  //  Typeset the given nodes
  //
  Typeset: function (update) {
    return MathJax.Hub.Typeset(update.nodes,{});
  },

  BeginMath: function () {
    //
    //  Save the start time for this paragraph
    //
    this.time = new Date().getTime();
  },
  BeginInput: function () {
    //
    //  Skip any paragraphs that aren't being updated, and
    //  update the equation numbers and macro definitions
    //  accordingly
    //
    var TEX = MathJax.InputJax.TeX, par;
    while (this.i < this.update[this.j]) {
      par = this.par[this.i++];
      this.eqNum += par.number;
      for (i = 0, m = par.defs.length; i < m; i++) {
        TEX.rootStack.Def.apply(TEX.rootStack,par.defs[i]);
      }
    }
    TEX.resetEquationNumbers(this.eqNum,true);
    //
    //  Store new macro and label definitions here
    //
    par = this.par[this.i];
    if (par) {
      if (!par.replaced) {par.olddefs = par.defs.all; par.oldlabels = par.labels.join('')}
      par.defs = []; par.defs.all = [];
      par.labels = [];
    }
  },
  TeXFilter: function (data) {
    //
    //  Get any new labels for this paragraph
    //
    var AMS = MathJax.Extension["TeX/AMSmath"];
    var labels = this.par[this.i].labels;
    for (var id in AMS.eqlabels) {if (AMS.eqlabels.hasOwnProperty(id)) {
      labels.push(id+"="+AMS.eqlabels[id])
    }}
  },
  TeXDef: function (def) {
    //
    //  Record the definition
    //
    var defs = this.par[this.i].defs;
    defs.push(def); defs.all.push(def[0]+"{"+def[1]+"}");
  },
  EndInput: function () {
    //
    //  Record the undefined references,
    //  the new definitions, and the equation number
    //  for this paragraph
    //
    var AMS = MathJax.Extension["TeX/AMSmath"];
    var par = this.par[this.i];
    if (par) {
      par.refs = AMS.refs; AMS.refs = [];
      par.defs.all = par.defs.all.join("");
      par.number = AMS.startNumber - this.eqNum;
      this.eqNum = AMS.startNumber;
      //
      //  If this is an update to an existing paragraph,
      //  check if it should force a refresh later
      //
      if (!par.replaced) {
        if (this.par.length !== this.update.length &&
            par.labels.join('') !== par.oldlabels) {this.refreshAll = true}
        if (this.update.length-this.j !== this.par.length-this.i &&
            par.defs.all !== par.olddefs) {this.refreshRest = true}
        delete par.olddefs; delete par.oldlabels; 
      }
    }
  },
  EndMath: function () {
    //
    //  Record the tie it took for this paragraph
    //  and go on to the next one.
    //
    var par = this.par[this.i];
    if (par) {
      var time = new Date().getTime();
      par.time = time - this.time; this.time = time;
      delete par.update; this.updateNeeded--;
      if (this.refreshRest || this.refreshAll) {MathJax.Hub.Cancel()}
    }
    this.j++; this.i++;
  },


  //
  //  Finish an incremental typesetting pass by checking if
  //  there are any undefined references that need updating,
  //  then checking if labels, defs, or equation numbers
  //  have changed, which would require updating additional
  //  paragraphs.
  //
  PostTypeset: function (update) {
    var incremental = this.incremental; this.incremental = false;
    if (MathJax.Hub.cancelTypeset && this.cancelled) return;
    //
    //  Check if there are undefined references that might have been
    //  defined in this update, and reprocess if so.
    //
    for (var i = 0, m = this.update.length; i < m; i++) {
      var par = this.par[this.update[i]];
      if (par.refs.length) {this.refs = this.refs.concat(par.refs); par.refs = []}
    }
    if (incremental && this.refs.length && !this.refreshAll && !this.refreshRest) {
      var queue = MathJax.Callback.Queue(
        ["Reprocess",MathJax.Hub,this.refs,{}],
        function () {if (!MathJax.Hub.cancelTypeset) {this.refs = []}}
      );
      return queue.Push(["PostTypeset",this,update]);
    }
    //
    //  Set the timer for the color removal
    //
    if (this.highlight && !this.ctimeout)
      {this.ctimeout = setTimeout(this.Unmark,this.classDelay)}
    //
    //  
    var labels = [], defs = [], number = 0;
    if (this.replace) {
      for (i = 0, m = this.update.length; i < m; i++) {
        var par = this.par[this.update[i]];
        if (par.replaced) {
          labels = labels.concat(par.labels.join(''));
          defs = defs.concat(par.defs.all);
          number += par.number;
          delete par.replaced;
        }
      }
      this.replace = false;
      this.loopCount = 0; // avoid any possibility of infinite loop
                          //  (shouldn't happen anyway, but I'm paranoid)
//console.log("--------");
//console.log(this.updateNeeded);
//console.log(this.oldlabels,labels);
//console.log(this.olddefs,defs);
//console.log(this.oldnumber,number);
    }
    if (update.nodes.length !== this.preview.childNodes.length) {
      if (this.refreshAll || labels.join('') !== this.oldlabels) {
        this.MarkForUpdate(0); this.refreshAll = this.refreshRest = false; this.refs = [];
      } else if (this.refreshRest || defs.join('') !== this.olddefs) {
        this.MarkForUpdate(this.i); this.refreshRest = false;
      } else if (number !== this.oldnumber) {
        this.MarkForUpdate(this.i,true);
      }
      if (this.updateNeeded && this.loopCount++ < 10) {
        var delay = Math.min(this.labelDelay,3*this.keyRate);
        if (this.getTime() < 2*this.keyrate) {this.Refresh()}
          else {this.ltimeout = setTimeout(this.Refresh,delay)}
      }
    }
  },

  //
  //  Mark the paragraphs from i onward for future updating
  //
  MarkForUpdate: function (i,numbered) {
    for (var m = this.par.length; i < m; i++) {
      if (!this.par[i].update && (!numbered || this.par[i].number)) {
        this.par[i].update = true; this.updateNeeded++;
      }
    }
  },
  //
  //  Get the list of nodes needing update,
  //  and their indices.
  //
  GetMarked: function () {
    var AMS = MathJax.Extension["TeX/AMSmath"];
    var nodes = [], indices = [], par = this.par;
    for (var i = 0, m = par.length; i < m; i++) {
      if (par[i].update) {
        var node = this.typeset.childNodes[i];
        nodes.push(node); indices.push(i);
        this.addChanged(node);
        for (var j = 0, n = par[i].labels.length; j < n; j++) {
          delete AMS.labels[par[i].labels[j].split(/=/)[0]];
        }
      }
    }
    return {nodes:nodes, indices:indices, replace:false};
  },
  
  //
  //  Remove the colors from the updated paragraphs
  //
  Unmark: function () {
    Preview.ctimeout = null; var nodes = Preview.typeset.childNodes;
    for (var i = 0, m = nodes.length; i < m; i++) {Preview.removeChanged(nodes[i])}
  },
  //
  //  Update the paragraphs that have pending updates
  //
  Refresh: function (method) {
    if (!method) {method = "Reprocess"}
    var update = Preview.GetMarked();
    Preview.ltimeout = null; this.replace = false;
    this.oldlabels = this.olddefs = ""; this.oldnumber = 0;
    if (update.nodes.length) {
      MathJax.Hub.Queue(
        ["PreTypeset",Preview,update],
        [method,MathJax.Hub,update.nodes,{}],
        ["PostTypeset",Preview,update]
      );
    }
  },

  //
  //  Force a complete refresh of the preview
  //
  Redraw: function () {
    this.typeset.innerHTML = this.preview.innerHTML;
    this.MarkForUpdate(0); this.updateNeeded = this.par.length;
    this.Refresh("Typeset");
  },

  //
  //  Find out how long updating the currently pending updates
  //  would be likely to take.
  //
  getTime: function () {
    var time = 0, i = 0, m = this.par.length;
    while (i < m) {if (this.par[i].update) {time += this.par[i].time}; i++}
    return time;
  },

  //
  //  Add the changed class to an element
  //
  addChanged: function (node) {
    if (!(node.className && node.className.toString().match(/(^| )changed( |$)/))) {
      if (node.className != "") {node.className += " changed"} else {node.className = "changed"}
    }
  },
  //
  //  Remove the "changed" class from an element (leaving all other classes)
  //
  removeChanged: function (node) {
    if (node.className) {
      node.className = node.className.toString()
                           .replace(/(^|\s+)changed(\s|$)/,"$2")
                           .replace(/^\s+/,"");
    }
  }

};

//
//  Hook into the math signals
//
MathJax.Hub.Register.MessageHook("Begin Math",function () {
  if (Preview.incremental) {Preview.BeginMath()}
});
MathJax.Hub.Register.MessageHook("End Math",function () {
  if (Preview.incremental) {Preview.EndMath()}
});
MathJax.Hub.Register.StartupHook("TeX Jax Ready",function () {
  MathJax.InputJax.TeX.postfilterHooks.Add(function (data) {
    if (Preview.incremental) {Preview.TeXFilter(data)}
  });
});
MathJax.Hub.Register.MessageHook("Begin Math Input",function () {
  if (Preview.incremental) {Preview.BeginInput()}
});
MathJax.Hub.Register.MessageHook("End Math Input",function () {
  if (Preview.incremental) {Preview.EndInput()}
},5); // priority = 5 to make sure it is before AMS.eqlabels are removed.

//
//  Hook into the definition routines to record
//    new definitions that occur.
//
MathJax.Hub.Register.StartupHook("TeX begingroup Ready",function () {
  var STACK = MathJax.InputJax.TeX.eqnStack;
  var DEF = STACK.Def;
  STACK.Def = function () {
    if (Preview.incremental) {Preview.TeXDef([].slice.call(arguments,0))}
    DEF.apply(this,arguments);
  }
  //
  //  Temporary hack to fix typo in begingroup.js
  //
  MathJax.InputJax.TeX.rootStack.stack[0].environments =
    MathJax.InputJax.TeX.Definitions.environment;
});

//
//  Set up MathJax to allow canceling of typesetting, if it
//    doesn't already have that.
//
(function () {
  var HUB = MathJax.Hub;

  HUB.Queue(function () {
    ready = true;
    HUB.processUpdateTime = 50;  // reduce update time so that we can cancel easier
    HUB.Config({
      "HTML-CSS": {EqnChunk: 10, EqnChunkFactor: 1}, // reduce chunk for more frequent updates
             SVG: {EqnChunk: 10, EqnChunkFactor: 1}
    });
  });

  if (!HUB.Cancel) {

    HUB.cancelTypeset = false;
    var CANCELMESSAGE = "MathJax Canceled";

    HUB.Register.StartupHook("HTML-CSS Jax Config",function () {
      var HTMLCSS = MathJax.OutputJax["HTML-CSS"], TRANSLATE = HTMLCSS.Translate;
      HTMLCSS.Augment({
        Translate: function (script,state) {
          if (HUB.cancelTypeset || state.cancelled) {throw Error(CANCELMESSAGE)}
          return TRANSLATE.call(HTMLCSS,script,state);
        }
      });
    });

    HUB.Register.StartupHook("SVG Jax Config",function () {
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<!--
 | There must be some missing constraints. If $\alpha_n$ is allowed to be negative, we get the following counterexample. $\smash{\rlap{\phantom{\Bigg(}}}$
 | 
 | Define
 | $$
 | u_{n+1}=(1-\alpha_n)u_n+\beta_n\tag{1}
 | $$
 | and
 | $$
 | A_n=\prod_{k=1}^{n-1}(1-\alpha_k)\tag{2}
 | $$
 | By induction, it can be verified that
 | $$
 | u_n=A_n\left(u_1+\sum_{k=1}^{n-1}\frac{\beta_k}{A_{k+1}}\right)\tag{3}
 | $$
 | For $j\ge1$, define
 | $$
 | n_j=\left\{\begin{array}{}
 | 2^{j(j-1)/2}&\text{when }j\text{ is odd}\\
 | 2^{j(j-1)/2+1}&\text{when }j\text{ is even}
 | \end{array}\right.\tag{4}
 | $$
 | and for $n\ge1$,
 | $$
 | \alpha_n=\left\{\begin{array}{}
 | \frac{1}{n+1}&\text{for }n_j\le n< n_{j+1}\text{ when }j\text{ is odd}\\
 | -\frac1n&\text{for }n_j\le n< n_{j+1}\text{ when }j\text{ is even}
 | \end{array}\right.\tag{5}
 | $$
 | Obviously, $\displaystyle\lim_{n\to\infty}\alpha_n=0$.
 | 
 | Using telescoping products, it is not difficult to show that
 | $$
 | \frac{A_{n_{j+1}}}{A_{n_j}}=\left\{\begin{array}{}
 | \frac{n_j}{n_{j+1}}=2^{-j-1}&\text{when }j\text{ is odd}\\
 | \frac{n_{j+1}}{n_j}=2^{j-1}&\text{when }j\text{ is even}
 | \end{array}\right.\tag{6}
 | $$
 | Equation $(6)$ yields
 | $$
 | A_{n_j}=\left\{\begin{array}{}
 | 2^{-(j-1)/2}&\text{when }j\text{ is odd}\\
 | 2^{-(3j-2)/2}&\text{when }j\text{ is even}
 | \end{array}\right.\tag{7}
 | $$
 | Furthermore, using the standard formula for the partial harmonic series, when $j$ is odd,
 | $$
 | \begin{align}
 | \sum_{n=n_j}^{n_{j+1}-1}\alpha_n
 | &=\log\left(\frac{n_{j+1}}{n_j}\right)+O\left(\frac{1}{n_j}\right)\\
 | &=(j+1)\log(2)+O\left(2^{-j(j-1)/2}\right)\tag{8}
 | \end{align}
 | $$
 | and when $j$ is even,
 | $$
 | \begin{align}
 | \sum_{n=n_j}^{n_{j+1}-1}\alpha_n
 | &=-\log\left(\frac{n_{j+1}}{n_j}\right)+O\left(\frac{1}{n_j}\right)\\
 | &=-(j-1)\log(2)+O\left(2^{-j(j-1)/2}\right)\tag{9}
 | \end{align}
 | $$
 | Combining $(8)$ and $(9)$ yields
 | $$
 | \sum_{n=1}^{n_j-1}\alpha_n=\left\{\begin{array}{}
 | \frac{j-1}{2}\log(2)+O(1)&\text{when }j\text{ is odd}\\
 | \frac{3j-2}{2}\log(2)+O(1)&\text{when }j\text{ is even}
 | \end{array}\right.\tag{10}
 | $$
 | Equation $(10)$ says that $\displaystyle\sum_{n=1}^\infty\alpha_n=\infty$.
 | 
 | Define
 | $$
 | \beta_n=\left\{\begin{array}{}
 | 2^{-j}&\text{when }n=n_j-1\text{ for }j\text{ even}\\
 | 0&\text{otherwise}
 | \end{array}\right.\tag{11}
 | $$
 | Summing the geometric series yields $\displaystyle\sum_{n=1}^\infty\beta_n=\frac13$.
 | 
 | Using $(3)$, we get
 | $$
 | \begin{align}
 | u_{n_{j+1}}
 | &=A_{n_{j+1}}\left(u_1+\sum_{k=1}^{n_{j+1}-1}\frac{\beta_k}{A_{k+1}}\right)\\
 | &\ge\frac{A_{n_{j+1}}}{A_{n_j}}\beta_{n_j-1}\\
 | &=2^{j-1}\cdot2^{-j}\\
 | &=\frac12\tag{12}
 | \end{align}
 | $$
 | when $j$ is even. $(12)$ says that $\displaystyle\lim_{n\to\infty}u_n\not=0$.
-->