I have before me the following sequence, recursively defined: $\\(a_n)_{n\in\mathbb N} \\a_1 = 1, a_{n+1} = \sqrt 1+a_n \ge 1$
I am to show the following: First, I ought to prove that $1\le a_n \le 2$ and that $a_n\le a_{n+1}$ I appreciate that this can be done throuh induction, not least because the first step here is basically given by the above definition of $a_1$, namely to show that this holds true for $a_1$ and then 'induce' the rest for $a_{n+1}$. My only issue here is that this latter part traps me a little.
Secondly, I ought to show that $(a_n)_{n\in\mathbb N}$ converges, which - well, I'll freely admit it, touches a nerve in the sense that while I understand the definitions etc. more or less intuitively, I still somewhat struggle with the actual applications of them, their existence 'in actu' such as here. Thus, any help with showing me exactly that this sequence converges is greatly appreciated.
Thanks everyone!
