$\sum_{n=1}^{\infty}a_{2n}$ converges Let $\sum_{n=1}^{\infty}a_n$ be a converging series with $a_n>0$,  $\forall n\in \mathbb N$. I want to prove that $\sum_{n=1}^{\infty}a_{2n}$ also converges. 
I know intuitively since all the terms in the sequence is positive, then the sum of $a_{2n}$ has to be less than the sum of $a_n$, but how would I show this rigorously? Or using a convergence test?
 A: Hint: What can you say about the convergence of the series with terms
$$b_k = \left\{ \begin{array}{cc}
   a_k & k \text{ even} \\
   0 & k \text{ odd} 
\end{array} \right.$$
perhaps using a comparison?
A: Consider the sequence of partial sums $$S_n=\sum_{k=1}^n a_k.$$ You know that this sequence converges, it is strictly monotone and bounded.
On the other hand, the sequence of partial sums for the second series satisfies
$$S'_n=\sum _{k=1}^{n}a_{2k}\le S_{2n}.$$
Therefore, the sequence $S'_n$ is strictly monotone, bounded, hence it converges. By definition, $\sum_{k\ge} a_{2k}$ converges.
A: Lemma: Suppose $c_1,c_2,c_3,\ldots\ge 0$.  Then
$$\lim_{n\to\infty} \sum_{k=1}^n c_k = \sup\left\{ \sum_{k\in A} c_k : A \text{ is a finite subset of }\{1,2,3,\ldots\} \right\}. $$
Lemma:
\begin{align}
& \left\{ \sum_{k\in A} c_k : A \text{ is a finite subset of }\{2,4,6,\ldots\} \right\} \\[10pt]
\subseteq & \left\{ \sum_{k\in A} c_k : A \text{ is a finite subset of }\{1,2,3,\ldots\} \right\}.
\end{align}
Lemma: Suppose $\mathcal T, S \subseteq \mathbb R$ and $\mathcal T\subseteq \mathcal S$.  Then $\sup\mathcal T \le \sup\mathcal S$.
The proof of the last lemma consists mostly of observing that every upper bound of $\mathcal S$ is an upper bound of $\mathcal T$.
See if you can prove the three lemmas and then use them to prove the proposition you want to prove.
