Convergence of $\sum\limits_{n=1} a_n$ implies convergence of $\sum\limits_{n=1} \left( a_{2n} + a_{2n+1} \right)$ I need to show that if the series $\sum\limits_{n=1} a_n$ converges then the series $\sum\limits_{n=1} \left( a_{2n} + a_{2n+1} \right)$ must also converge. Does it suffice to argue that since the two sums differ only by a single term $a_1$ and $a_n \in \mathbb{R}$ for all $n$ (since the first series converges), then the second must also converge? That is, since
$$
\sum_{n=1} a_n = a_1 + a_2 + \cdots
$$
and
$$
\sum_{n=1} \left( a_{2n} + a_{2n+1} \right) = (a_2 + a_3) + (a_4 + a_5) + \cdots
$$
we can write
$$
\sum_{n=1} a_n = a_1 + \sum_{n=1} \left( a_{2n} + a_{2n+1} \right)
$$
and so clearly if the left sum converges, the right must as well.
Does this seem like a sufficiently rigorous argument for an upper-division real analysis course? My class is using "Elementary Analysis: The Theory of Calculus," by the way.

Hint: The sequence of partial sums of
$$
\sum_{n=1}^\infty a_{2n}+a_{2n+1}
$$
is a subsequence of the sequence of partial sums of
$$
\sum_{n=2}^\infty a_n
$$
