JQuery Search Text and UI functionality

I was able to create a search box but as the user types, I wanted the user the ability to narrow down their search.  So let's say that when the user starts typing, there are 100 records to be displayed, if a user types 3 or 4 letters to narrow down their search, I want only the filtered text to appear so maybe there would only be 10 or so records.  Below is my code on pulling all of the records but as I type, the list doesn't change nor does it narrow down.  How would I do this?  Please view code below:

<!DOCTYPE html>
<head>
    <script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
    <script src="http://code.jquery.com/ui/1.10.3/jquery-ui.js"></script>
    <link href="http://ajax.googleapis.com/ajax/libs/jqueryui/1.8/themes/base/jquery-ui.css" rel="stylesheet" type="text/css" />
    <title></title>
    <meta charset="utf-8" />   
</head>
<body>
<div class="ui-widget">
    <form>
        <input type="text" id="search" name="search" />
    </form>
</div>
 <script type="text/javascript">
    
     $('#search').autocomplete({
         source: function (request, response) {
             $.ajax({
                 url: "http://195.129.8.190/Service/rest/Services/v1/departments",
                 dataType: "json",
                 data: { term: request.term },
                 success: function (data) {
                     response($.map(data, function (item) {
                         return {
                             label: item.deptName,
                             id: item.deptId
                         };
                     }));
                 }
             });
         },
         minLength: 3,
         select: function (event, ui) {
             $('#search').val(ui.item.deptName);
             
         }
     });
 </script>

</body>
</html>

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VBBRettAsked:
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kblauCommented:
<!DOCTYPE html>
<html>
<head>
    @*credit to http://kilianvalkhof.com/2010/javascript/how-to-build-a-fast-simple-list-filter-with-jquery/*@
    <title>Simple Map</title>
    <meta name="viewport" content="initial-scale=1.0">
    <meta charset="utf-8">
    <script src="~/Scripts/jquery-1.9.1.min.js"></script>
    <script type="text/javascript">
        (function ($) {
            // custom css expression for a case-insensitive contains()
            jQuery.expr[':'].Contains = function (a, i, m) {
                return (a.textContent || a.innerText || "").toUpperCase().indexOf(m[3].toUpperCase()) >= 0;
            };


            function listFilter(header, list) { // header is any element, list is an unordered list
                // create and add the filter form to the header
                var form = $("<form>").attr({ "class": "filterform", "action": "#" }),
                    input = $("<input>").attr({ "class": "filterinput", "type": "text" });
                $(form).append(input).appendTo(header);

                $(input)
                  .change(function () {
                      var filter = $(this).val();
                      if (filter) {
                          // this finds all links in a list that contain the input,
                          // and hide the ones not containing the input while showing the ones that do
                          $(list).find("a:not(:Contains(" + filter + "))").parent().slideUp();
                          $(list).find("a:Contains(" + filter + ")").parent().slideDown();
                      } else {
                          $(list).find("li").slideDown();
                      }
                      return false;
                  })
                .keyup(function () {
                    // fire the above change event after every letter
                    $(this).change();
                });
            }


            //ondomready
            $(function () {
                listFilter($("#header"), $("#list"));
            });
        }(jQuery));
    
    </script>
    <style>
        body {
            background: #fff; 
            font-size:13px;
        }

        #wrap {
            position: relative;
            width: 220px;
            margin: 50px auto 0;
        }

        #header {
            position: relative;
            line-height: 1em;
        }

        .filterform {
            width: 220px;
            font-size: 12px;
            display: block;
        }

            .filterform input {
                -moz-border-radius: 5px;
                border-radius: 5px;
                -webkit-border-radius: 5px;
            }
    </style>
</head>
<body>
    <h1 id="header">List of countries</h1>

    <ul id="list">
        <li><a href="#/australia/">Australia</a></li>
        <li><a href="#/austria/">Austria</a></li>
        <li><a href="#/belgium/">Belgium</a></li>
        <li><a href="#/brazil/">Brazil</a></li>
        <li><a href="#/canada/">Canada</a></li>
        <li><a href="#/denmark/">Denmark</a></li>
        <li><a href="#/finland/">Finland</a></li>
        <li><a href="#/france/">France</a></li>
        <li><a href="#/germany/">Germany</a></li>
        <li><a href="#/greece/">Greece</a></li>
        <li><a href="#/ireland/">Ireland</a></li>
        <li><a href="#/israel/">Israel</a></li>
        <li><a href="#/italy/">Italy</a></li>
        <li><a href="#/japan/">Japan</a></li>
        <li><a href="#/luxembourg/">Luxembourg</a></li>
        <li><a href="#/mexico/">Mexico</a></li>
        <li><a href="#/netherlands/">Netherlands</a></li>
        <li><a href="#/norway/">Norway</a></li>
        <li><a href="#/poland/">Poland</a></li>
        <li><a href="#/portugal/">Portugal</a></li>
        <li><a href="#/russia/">Russia</a></li>
        <li><a href="#/spain/">Spain</a></li>
        <li><a href="#/sweden/">Sweden</a></li>
        <li><a href="#/switzerland/">Switzerland</a></li>
        <li><a href="#/turkey/">Turkey</a></li>
        <li><a href="#/united-kingdom/">United Kingdom</a></li>
        <li><a href="#/united-states/">United States</a></li>
     </ul>
  
</body>
</html>

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VBBRettAuthor Commented:
Thank you for the solution.
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