Show sequence convergence

1. Let $\displaystyle (x_n)$ be a sequence with $\displaystyle x_1 = 1$ and $\displaystyle x_n = x_{n-1} - \frac{(-1)^n}{n}$. Prove convergence.

I tried to prove this is cauchy, but couldn't make much progress.

2. Let $\displaystyle (s_n)$ be a sequence with $\displaystyle s_1 = 3$, $\displaystyle s_{n+1} = \frac{2}{3}s_n + \frac{4}{3s_n}$. Prove convergence and evaluate the limit.

I know the limit is 2, but I don't understand how to show this using an $\displaystyle \epsilon$-proof.