It is valid (if ) but I don't understand why you need it. Is it to say that if then ?The problem that im having is with my first line, there exists N s.t. n+k>n>N. Is this a valid statement? I keep getting the feeling that there is something wrong with it.
Let's assume that converges, that is to say that hence by taking we get that hence converges.
It's exactly the idea you used but the proof is both shorter and clearer.
Hope that helps.