Prove that converges if and only if converges for any

My approach..

Assume that the latter series converges then this inequality holds for some real

for any

And thus does also the first series converge. They only differ by a constant depending on .

Now assume that diverges to for some and that converges.

This inequality holds for some

for any

But this inequality doesn't hold for any

for any

Contradiction!

My assumtition must be wrong.

Either does both diverge to or converge.

Is there something wrong? Something missing to complete the proof?