be the sequence $a_{n}= \sqrt{2},\sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}},...$ Then you have to first prove that it converges, and then prove its limit. Also the problem gives a suggestion to use that: $(a_{n+1})^2=2a_{n}$ What I have to the moment is that the sequence is crescent and 2 is an upper bound, so I think its limit is 2, but I do not know how to prove it. also, $a_{n}=\prod_{i=1}^{n}{2^{1/2^i}}$ if it helps.
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A sequence which is increasing and upper bounded converges. Let $L = \lim_n a_n$. Note that $\lim_na_{n+1}=L$. Now use the fact that $(a_{n+1})^2=2a_n$ to find $L$. – azif00 Apr 19 '24 at 01:07
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You can use induction to prove 2 is an upper bound of the sequence. Then the sequence is obviously increasing. – Gyh Apr 19 '24 at 01:12