#1

Prove the sequence $\displaystyle \{x_n\}$ defined in the following is increasing and bounded.

$\displaystyle x_n=1+\frac{x_{n-1}}{1+x_{n-1}}$

I suppose to use MI to prove and let S(n) by the statement

$\displaystyle x_{n+1}\geq x_{n}$ and $\displaystyle 1\leq x_n<2$but i cant prove the sequence is increasing for the case n=k+1, am i starting with wrong statement ?

#2

Set $\displaystyle a_1=1$ ,and for $\displaystyle n\geq2$, $\displaystyle a_{n+1}=\frac{1}{2+a_n}$.Show that the sequence $\displaystyle \{a_n\}$ is convergent.

I am confusing how to let the statement.