Because x1= √ 2x2 = √ (2+√ 2) > √ 2 = x1

So x2>x 1

Suppose that when n=k, there is xk >；; x(k- 1)

Then when n=k+ 1, x (k+1)-xk = √ (2+xk)-√ (2+x (k-1)) = [xk-x (k-1]

By assuming xk & gtX(k- 1), X (k+1)-XK >; 0

X (k+ 1) >: xk

So this series is monotonically increasing.

2. Prove that this sequence is bounded by mathematical induction.

Because x 1 = √ 2

Suppose when n=k, there is xk.

Then when n=k+ 1, x (k+ 1) = √ (2+xk) < √(2+2)=2.

X (k+ 1)

So the upper bound of this series is 2.