Let $(x_n)$ be a decreasing sequence in R with $x_n \geq 0$ for all $n$ and $\lim_{n\to\infty} x_{n} = 0$.
Define: $s_k = x_1 - x_2 + ... + (-1)^{k+1}x_k$.
Prove that $x_k$ is a Cauchy sequence thus proving that $(s_k)$ has a limit.
I'm not sure how to start this problem at all.
You have to prove that $| s_m - s_n |$ is definitely less than $\epsilon$. Suppose $m>n$: then $$| s_m - s_n | = |x_1 - x_2 + \dots (-1)^{m+1}x_m - (x_1 - x_2 + \dots (-1)^{n+1}x_n) | = $$ $$|(-1)^{n+2}x_{n+1} + (-1)^{n+3}x_{n+2} + \dots (-1)^{m+1}x_m | \leq | x_{n+1} | < \epsilon. $$ To see that $$|(-1)^{n+2}x_{n+1} + (-1)^{n+3}x_{n+2} + \dots (-1)^{m+1}x_m | \leq | x_{n+1} | $$ just note that $x_n$ is positive and decreasing. Now $$ | x_{n+1} | < \epsilon $$ because $x_n$ converges to zero