The sequence is recursively defined by a1 = √(2), an+1 = √(2+an) for n ≥ 1.
The sequence {an} is bounded above. Call this upper bound L.
Show {an} is monotone increasing. Thus {an} has a limit. Why?
Find this limit?
Thank you sooo much for all the help guys!!!
You have by definition that a number L is the limit of a sequence iff
A sequence converges iff it has a limit.
You can see from the definition that every monotone bounded sequence has a limit and thus converges.
So your sequence being both monotone and bounded, necessarily converges and thus has a limit.
flyingsquirrel means generally they need not be the same one. Since what I said could be misleading. You might think it works for any function.
However for , we can limit both sides to infinity to get
Since you have already proved that has a limit. Call it l. Its easy to see that . So you can continue like what I did before.