Prove that the following sequence is convergent: `u_1=sqrt(2),` `u_2=sqrt(2+sqrt(2)), ldots, u_n=sqrt(2+sqrt(2+sqrt(2+sqrt(2+cdots+sqrt(2+sqrt(2))))))`
In order to prove that your sequence is convergent we will use the following theorem:
A sequence of real numbers is convergent if it is monotone and bounded.
1. proof of monotonicity
We will prove this by induction:
i) `x_2=sqrt(2+sqrt(2))>sqrt(2)=x_1` which proves the base.
ii) Assume that for every `k leq n` `x_k>x_(k-1).`
iii) Now we use assumption for `k=n` and add number 2 to each side.
(The entire section contains 225 words.)
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By proof of monotonicity why did you add 2 at both sides,and is there any other way of proving this which is more compactible.