Call functions from function inside an object (object literal)

You can try the following code. It should work:

var RunApp = {

  init: function(){   
     RunApp.run()
  },

  run: function() { 
     alert("It's running!");
  }
};

That code is only a declaration. You need to actually call the function:

RunApp.init();

Demo: http://jsfiddle.net/mattball/s6MJ5/

There is nothing magical about the init property of an object, which you happen to have assigned a function to. So if you don't call it, then it won't run. No functions are ever executed for you when constructing an object literal like this.

As such, your code becomes this:

var RunApp = {
    init: function(){   
         this.run()
    },
    run: function() { 
         alert("It's running!");
    }
};

// Now we call init
RunApp.init();