If $(a_{n_k}) $is a subsequence of $(a_n) $, show that $\sum a_n $ is absolutely convergent iff every subseries of it convergent. If $(a_{n_k})$ is a subsequence of $(a_n)$, show that $\sum a_n $is absolutely convergent  iff every subseries of it convergent.
My try...If the series is absolutely convergent, then the partial sum sequence  say, $(s_n),s_n=\sum |a_n|$ is a cauchy sequence. Hence for any $r>0$ there exists a natural number $m $such that $|a_m|+|a_{m+1}|+|a_{m+2}|+\cdots<r $. Now any subsequence is constructed from these $|a_{m+p}|$ terms.For infinitely many $p\in N $.
But I can't prove the converse part.From my proof it seems to me that all the subseries are absolutely convergent if the 
given series is absolutely convergent. Am I correct? 
 A: If $\sum_n |a_n|$ converges, then for any subsequence $a_{n_k}$, $\sum_k |a_{n_k}|\leq\sum_n |a_n|<\infty$.
Suppose all the $a_i$ are real. Without loss of generality, you may also assume they're all non-zero. Assume that for any subsequence $a_{n_k}$, $\sum_k a_{n_k}$ converges. This is the case for the subsequence $a_{n_k}$ of positive terms and the subsequence $a_{m_k}$ of negative terms. Define $b_j=\max(a_j,0)$ and $c_j=-\min(a_j,0)$, so that $\sum_j b_{j} = \sum_k a_{n_k}$ and $\sum_j c_{j} = \sum_k a_{m_k}$.  Then $$\sum_k a_{n_k} - \sum_{k} a_{m_k} = \sum_j (b_{j} - c_{j}) = \sum_j |a_j|$$
Drop the requirement that the $a_i$ are real. Note that for any subsequence $a_{n_k}$, $\sum_k Re(a_{n_k})$ and $\sum_k Im(a_{n_k})$ converge (because $\sum_k a_{n_k}$ converges). Using the statement proved for real numbers, we have $\sum_k |Re(a_{n_k})| <\infty $ and $\sum_k |Im(a_{n_k})| < \infty$.
Since $|a_{n_k}|\leq |Re(a_{n_k})| + |Im(a_{n_k})| $, $\sum_k |a_{n_k}| < \infty$.
A: If a series $\sum_{n=1}^\infty a_n$ fails to be absolutely convergent then $\sum_{n=1}^\infty |a_n| = +\infty,$ and so either the sum of the positive terms is $+\infty$ or the sum of the negative terms is $-\infty$ or both. Thus either the subseries containing only positive terms or the one containing only negative terms fails to converge.
Now assume the series converges absolutely. Then the series containing only positive terms converges and so does the one containing only negative terms, since the only way for a series in which all terms have the same sign can diverge is by diverging to infinity. So consider an arbitrary subseries. Split it into a part containing only nonnegative terms and a part containing only negative terms. Both converge because a subseries of a convergent series in which all terms have the same sign converges. Finally, the sum of two convergent series converges.
