consider $ \displaystyle \sum_{n=1}^\infty (-1)^{n-1}a_n $ where $ (a_n) $ is a monotone decreasing sequence of nonnegative numbers with $ a_n \rightarrow 0 $ by the alternating series test, series of this form always converge.
Show that $ 0 \leq \displaystyle\sum_{n=1}^\infty (-1)^{n-1}a_n \leq a_1 $
There is a hint in the question:
if $ (S_N)$ denotes the sequence of partial sums, consider the subsequences $ (S_{2N}) $ and $ (S_{2N-1}) $ can you show that one is decreasing while the other is increasing?
I have proven the hint - however I am unable to proceed from here - Could someone please direct me to the correct direction?