Let $\sum a_{k}$ be a convergent series in a Banach space $E$. Show that $\lim_{n \to \infty} r_n =0$

Question

I'm doing Problem II.7.4 in textbook Analysis I by Amann/Escher.

My attempt:

Since $\sum a_k$ is convergent, let $a := \sum a_k$. We have $r_n = \sum_{k=n}^\infty a_k = \sum_{k=0}^\infty a_k - \sum_{k=0}^{n-1} a_k =$ $a - \sum_{k=0}^{n-1} a_k$. Take the limit of both sides, we have $$\lim_{n \to \infty} r_n = \lim_{n \to \infty} \left(a - \sum_{k=0}^{n-1} a_k \right) = a- \lim_{n \to \infty} \sum_{k=0}^{n-1} a_k = a-a =0$$

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My questions:

1. My proof is quite short. Is it fine or contains logical gaps/errors?

2. I don't see the need for the hypothesis that $E$ is Banach space. I would like to ask if it is actually needed.

Answer 1

Your proof is fine and it works on any normed space. That is, the hypothesis that your space is a Banach space is not needed.