# Prove that if the sequence of partial sums $A_n$ of $\sum a_n$ is bounded, then ${a_n}$ is bounded.

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?

• Proof is good, and written well. – dxiv May 8 '17 at 23:25
• Corrected some terminological inaccuracies. – avs May 8 '17 at 23:36