Can Herschfeld's Convergence Theorem be used to prove the convergence of a_n?

[Icebreaker]

"Let k\in \mathbb{N} and a_0=k. Let a_n=\sqrt{k+a_{n-1}}, \forall n\geq1 Prove that a_n converges."

If we look at the similar sequence b_0 = k and b_n = sqrt(a_n-1), then that sequence obviously converges to 1. Unfortunately, b_n<a_n so I can't use the squeeze theorem.

Any hints would be nice.

 

[1800bigk]

I would say let lim an = s also lim an-1 would still be s so you can use the limit properties and can get a quadratic with s^2 - s -k=0 you should be able to go from there

 

[devious_]

I would say let lim an = s also lim an-1 would still be s so you can use the limit properties and can get a quadratic with s^2 - s -k=0 you should be able to go from there

But you don't know if \lim a_n exists.

Have you tried checking if a_n is monotonic & bounded?

 

[siddharth]

You should first try to prove that the sequence is bounded.
Then if you show that it monotonically increases or decreases, you can prove that the sequence is convergent.

 

[Icebreaker]

It can easily be shown that it's monotonically increasing. However, it's the bounded part that gets me. Maybe I can use Herschfeld's Convergence Theorem?