Iam trying to display the selected list of records into the data table. there is no error invoked but still the data is not visible...
here is the code

<apex:page standardController="Product__c" extensions="prosearch">
  <apex:form >
  <apex:pageblock >
   <apex:pageBlockSection >
  
   <apex:pageBlockSectionItem >
   <apex:outputLabel >Select Department</apex:outputLabel>
      <apex:selectList value="{!selecteddept}" size="1">
           <apex:selectOptions value="{!deptitems}" rendered="true" />
      </apex:selectList>
   </apex:pageBlockSectionItem>
        <apex:commandButton id="search" Value="search" action="{!search}"/>

  
    <apex:outputPanel >
    <apex:outputtext value="{!selecteddept}"></apex:outputtext>
      <apex:datatable value="{!searchlist}" var="a" id="theTable" title="Iam HERE" >

            <apex:facet name="retrieved">We Have The Following Products</apex:facet>
       
       <apex:column >
            <apex:facet name="ProductName">Product Name</apex:facet>
            
            <apex:outputtext value="{!a.Product_Name__c}"/>
        </apex:column>
      
        <apex:column >
            <apex:facet name="ProductPrice">Product Price</apex:facet>
            <apex:outputText value="{!a.Price__c}"/>
        </apex:column>
      </apex:datatable>
    </apex:outputPanel>
   </apex:pageBlockSection>
 
  </apex:pageBlock>
 
  </apex:form>
</apex:page>


here is the apex code

public with sharing class prosearch {

    public prosearch(ApexPages.StandardController controller) {
     
    }
    public List<Product__c> searchlist{set; get;}
    public string selectedDept{set;get;}
    public string productname{ set; get;}
    public decimal productprice{ set;get;}
    public  string productimage{ set;get;}
    
    public string getselectedDept(){
      
       return selecteddept;
       
       }
        public void setselecteddept() {
           this.selecteddept = selecteddept;
    }
   public string deptItems{set; }

   public List<SelectOption> getdeptItems() {
            List<SelectOption> options = new List<SelectOption>();
             Schema.DescribeFieldResult fieldResult =Product__c.Product_department__c.getDescribe();   
        List<Schema.picklistEntry> ple = fieldResult.getPicklistValues();
          for(Schema.picklistEntry f:ple)   
        {   
            options.add(new selectOption(f.getLabel(),f.getValue()));                   
        }   
        return Options;
       
       
}


    public pagereference search(){
      return null;
      }
 
public List<Product__c> searchproduct(){


List<Product__c> searchlist = new List<Product__c>();
        searchlist = [select Product_Name__c,Price__c,Product_Image__c from Product__C where product_department__C =: selecteddept];
  
     return searchlist; 
     
 
    
 

}
}