Prove that if the sequence a(n+k) converges to l then the sequence a(n) also converges to l.
My proof basically looks like this:
there exists N s.t. n+k>n>N.
Assume the sequence a(n+k) converges to l and that the sequence a(n) diverges. By the definition: there exists L>0 s.t. for every K>0, there exists N that is an integer s.t n which is also an integer n>N -> |a(n)-L|<K
This means that if the sequence a(n) converges, then the sequence a(n+k) also converges because n+k>n>N. But if a(n) diverges, by the definition of divergence (eg not(definition of convergence)) a(n+k) also diverges. And since a(n+k) only converges when a(n) converges, there is a contradiction with my initial assumption where a(n+k) converges and a(n) diverges.
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.