Q:

Prove $\sum a_n+2a_{n+1}-a_{n+2}$ converges if and only if $\sum a_n$ converges

Let $\{a_n\}\subseteq \Bbb R$ be a sequence such that $a_n\to 0$.
Prove that $\sum a_n+2a_{n+1}-a_{n+2}$ converges if and only if $\sum a_n$ converges.

I'll post my attempt as an answer.

A:

If $\sum a_n$ converges, then obviously the other sum converges.
Now, suppose $c_N=\sum_{n=0}^N(a_n+2a_{n+1}-a_{n+2})$ converges. We can rewrite this as 
$$
c_N=\sum (-(a_{n+2}-a_{n+1})+(a_{n+1}-a_n)+2a_n)
$$
The first two terms are telescoping sums, so we get
$$
c_N = -a_{N+2}+a_1+a_{N+1}-a_0+\sum_{n=0}^N2a_n
$$
As $c_N$ converges and $a_N\to 0$, let $N\to +\infty$ and we get $c=a_1-a_0+\sum_0^\infty 2a_n$
I think (if everything before is correct), that I'm done here: we see that $\sum 2a_n$ converges and I believe I can take the $2$ out and say that $\sum a_n $ converges, right?

