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Prove that if the sequence of partial sums $A_n$ of $\sum a_n$ is bounded, then the sequence $(a_n)_{n \geq 1}$ is bounded.

Proof

Since the sequence $(A_n)_{n \geq 1}$ is bounded, there exists (by definition of boundedness) a real number $M$ such that $|A_n|\leq M$ for all $n~\in~\mathbb{N}$. Consider $|A_{n+1}-A_n|=|a_{n+1}|$. We have $$2M \geq |A_{n+1}|+|A_{n}| \geq |A_{n+1}-A_{n}| = |a_{n+1}|,$$ so that the sequence $(a_{n+1})_{n \geq 1}$ is bounded by $2M$. This completes the proof.

Question

I am sometimes not sure whether my proofs are totally correct. Did I make any logical errors in this proof?

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    $\begingroup$ Proof is good, and written well. $\endgroup$ – dxiv May 8 '17 at 23:25
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    $\begingroup$ Corrected some terminological inaccuracies. $\endgroup$ – avs May 8 '17 at 23:36
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Proof is good, and written well. – dxiv 12 mins ago

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