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If $\{\sum_{n=1}^{\infty} a_n\}$ is absolutely convergent if and only if the sequence of partial sums $\{S_n\}=\sum_{k=1}^{n} a_n$ is bounded

By theorem: Absolute Convergence implies Convergence.

Then if $\sum a_n$ converges then the sequence of partial sums $\{S_n\}=\sum_{k=1}^{n} a_n$ is a convergent sequence, and convergent sequences are bounded by by theorem: Convergent sequences are bounded.

Now suppose that the sequence of partial sums $\{S_n\}$ is bounded. Since the $a_n$ are all nonnegative, the sequence $\{S_n\}$ is monotone increasing, and all bounded monotone sequence converge by theorem: All bounded monotone sequences converge. Hence $\{S_n\}$ converges and so does the series.

I feel lost, am I right?

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    Are $a_n$'s non-negative? – 2017-02-09
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    Your statement is false. Every convergent sequence has bounded partial sums, but not every sequence with bounded partial sumis is convergent. Every absolutely convergent sequence is convergent, but not every convergent sequence is absolutely convergent. – 2017-02-09
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    @ArunBadajena in my exercise I don't know is $a_n$ is non-negative – 2017-02-09

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At the end of your proof you told us that $a_n \ge 0$ for all $n$. A little bit late..... .

If all $a_n \ge 0$, then absolute convergence and convergence is the same !

Then your statement reads:

$\sum_{n=1}^{\infty} a_n$ converges $\iff$ the sequence $(\sum_{k=1}^{n} a_n)_n$ is bounded.

Your proof for this statement is o.k.