For the forward implication I would break the proof into two parts.

First show that

I assume that you mean the above if k=0 the sequence is constant and equal to 1.

So suppose not( assume k < 0) and use what you have above to get a contradiction. Hint the sequence will be unbounded.

Then after you know that k > 0 you can make a direct argument to show that it converges to zero.

The reverse <= implication is not bad at all.