Prove the "Alternating Series Test"; .ie suppose that (x_n) is a positive
decreasing sequence with lim (x_n)= 0. Show that the alternating series
is convergent and the sum satisfies |s-s_k|≤ s_k
where s_k is the partial sum and s = lim (s_k).
Hint: Start by showing that (s_2k) and (s_2k+1) are both monotone.
I showed that the odd partial sums are monotone increasing and the even partial sums are monotone decreasing, but I couldn't find the rest.
If you help me, I will be really happy.