1. ## Sequence Convergence

so the sequence is

{ $\sqrt2, \sqrt(2+\sqrt2),\sqrt(2+\sqrt(2+\sqrt2), ...$ }

i need to prove by induction that the sequence is monotonic and bounded.
i can see how its monotonic, but how do i prove its bounded? and what is induction?

2. Originally Posted by ixion124
so the sequence is

{ $\sqrt2, \sqrt(2+\sqrt2),\sqrt(2+\sqrt(2+\sqrt2), ...$ }

i need to prove by induction that the sequence is monotonic and bounded.
i can see how its monotonic, but how do i prove its bounded? and what is induction?
Work with $s_1=\sqrt2$, $n\ge2,~s_n=\sqrt{2+s_{n-1}}$.
Use induction.

3. thanks!
but how would i prove it with induction?

4. Show that $s_1\le s_2\le 2$.

Show that if $s_N\le s_{N+1}\le 2$ then $s_{N+1}\le s_{N+2}\le 2$.