# Limit of Recursive sequences

Given

$Z_1 > 0$ and $a > 0$

$Z_{n+1} = (a+Z_n)^{1/2}$

To show $Z_n$ is convergent and find its limit.

The general approach would be to use PMI(Induction) to show that there exists an upper bound. And that the sequence $(Z_n)$ is monotone. And then using Monotone Convergence Theorem. However I am having great difficulty proving boundedness, reason being $Z_1$ is not a fixed number.

Monotone increasing could be proven for $n>=2$( please do tell me whether this approach is correct).

But boundedness ? Do I have to solve this problem using some other technique ?

Another such example would be

$Y_1 = (p)^{1/2}$ and $p>0$

$Y_{n+1} = (p+Y_n)^{1/2}$

I could solve this question without any problem, knowing the first term of the sequence.

You do know the first term of the sequence. It's $Z_1$. The problem says that it's given (and positive).