How do I prove that this recursive sequence converges?

Question

Let $a_1=\sqrt2$ and let $a_n=\sqrt{2+a_{n-1}}$ for $n \ge 2$. How do I prove that this sequence converges?

Visit this page for information on how to type with $\LaTeX$ and MathJax on this site so you can type things like $a_n = \sqrt{2+a_{n-1}}$ — JMoravitz, Dec 15 '16 at 21:35
It it converges to $L$, which natural relationship is "fulfilled" by $L$ ? — Jean Marie, Dec 15 '16 at 21:38
I'm sorry. I didn't know someone else had already asked this question. — wastewhite, Dec 15 '16 at 21:58
You might also check the question linked to the one suggested as a duplicate. For example. You can finds several posts specifically about $c=2$, like this one. — Martin Sleziak, Dec 15 '16 at 22:26

score 1 · Answer 1 · answered Dec 15 '16 at 21:37

1

Hints:

$$a_n=\sqrt{2+a_{n-1}}\le\sqrt{2+a_n}=a_{n+1}$$

$$a_n\le 2\iff \sqrt{2+a_{n-1}}\le\sqrt{2+2}=2$$

Can you see now how to prove by induction th above?

answered Dec 15 '16 at 21:37

DonAntonio

1 Answers1