If $\sum a_n$ is convergent and $a_n>0$, then $\sum (-1)^n a_n$ is convergent. 
If $\sum a_n$ is convergent and $a_n>0$, then $\sum (-1)^n a_n$ is convergent.

So far I've tried convergence/divergence tests and also I tried to prove this using partial sums. But tests do not work because the latter series includes negative terms. Maybe the Alternating Series Test could have worked but "$a_n$ is decreasing" is not an initial condition. I know the proof will be very easy but I am stuck here. Could you provide me with a little hint?

HINT: any absolutely convergent series is convergent.Let me know if you want the full solution
