I have one picklist with me, and i am trying to get the selected value of the picklist. below is the code

VF page :

                               <apex:outputlabel value="Select City" for="values" />

                                <apex:actionSupport event="onchange" action="{!getcitylist}" rerender="txt_city"/>                                   
                                 <apex:selectList value="{!cities}" size="1">
                                      <apex:selectOptions value="{!showcities}"/>
                                 </apex:selectList>

                                  <apex:inputText id="txt_city"/>

Apex Code :

      String sel_city;

      public String cities
        {
            get {return sel_city;}
            set {sel_city=value;}
        }

       public List<selectOption> getshowcities()
    {

            List<selectOption> Selectcity = new List<selectOption>();

              List<Regstr__c> citylist = new List<Regstr__c>();
              courselist = [Select  id, city_name__c FROM Regstr__c ];

              SelectCourse.add(new SelectOption('0','Select city'));

              for (Integer j=0;j<citylist .size();j++)
              {

                 
                  Selectcity .add(new SelectOption(citylist [j].id,citylist [j].city_name__c));
                 
              }      
            return Selectcity ;

    }  

   public void getcitylist()  //this function is called onchange event of dropdown

   {

         //once the selected item in the city drop down is changes, here i have to save the selected value of "city"dropdown  list in textbox "txt_city"

   }

I am new to salesforce, please suggest me how to write the scipt to save the value of dropdown in textbox.

Thanks in advance!!!!!!!!!!!!!!!!!!!!!!