Let $a_n \in \mathbb R$, such that $\sum_{ n=1}^{\infty }|a_n| = \infty $ and $\sum_{ n=1}^{m }a_n \ \rightarrow a\in \mathbb R$ as $m \rightarrow \infty$. Let $a_n ^+= \max\{a_n,0\}.$
Show that $\sum_{ n=1}^{\infty }a_n^+ = \infty $.
My attempts : I was thinking like this
$$|a_n|=a_n^{+}- a_n^{-} $$ $$\sum_{ n=1}^{\infty }|a_n|=\sum_{ n=1}^{∞ }a_n^{+} -\sum_{ n=1}^{\infty }a_n^{-}$$
After that I can't proceed further.
Please help me and tell me the solution or any hints can be appreciated.
Thanks.
Your problem can be restated as finding a sum of all positive elements of the series. The series is convergent, but not absolutely convergent, so its positive elements alone and negative elements alone produce divergent series.