Let $(x_n)$ be a sequence in a Banach space $X$. Which of the following conditions ensure(s) that $(x_n)$ is convergent in $X$?

Question

Let $(x_n)$ be a sequence in a Banach space $X$. Which of the following conditions ensure(s) that $(x_n)$ is convergent in $X$?

(a) $\|x_n − x_{n+1}\| \to 0.$

(b) $\sum^{\infty}_{n=1} \|x_n − x_{n+1}\|<\infty$

I think that the first condition is insuffuicent.

What if $x_n=\sum^{n}_{i=1}\frac{1}{i}?$.

What about the second one? I couldn't say anything about (b).

Edit 1

I think (b) works. we are supposed to see what about $\|x_m-x_n\|\leq\|x_m-x_{m+1}\|+\|x_{m+1}-x_{m+2}\|+ \dots +\|x_{n-1}-x_n\|$

Answer 1

The second condition implies that the sequence is convergent since it is a Cauchy sequence:

$\|x_{n+p}-x_n\|\leq \|x_{n+p}-x_{n+p-1}\|+....+\|x_{n+1}-x_n\|$ and since $\sum \|x_{n+1}-x_n\|<\infty$, for every $c>0$ there exists an integer $N$, such that for every $n>N, \|x_{n+p}-x_{n+p-1}\|+...+\|x_{n+1}-x_n\|<c$.