Can I convince the investors by presenting the idea and my skills without a working prototype?
A helpful way of looking at it might be that investors don't care much about the idea and the first order skill they care about is the ability to build a business that can grow big. That means that the most convincing thing is likely to be a quickly growing business. Then comes a product that people are using. Then a process in execution to get to a product that people are using. A working prototype can be part of that process or not (or rather the definition of 'working' is not a technical one but a commercial one in that people are using it and providing feedback.
An article I like: http://www.themacro.com/articles/2016/01/minimum-viable-prod...