Banach space divergent sequence.

Let $X$ be a real Banach space, and let $\{ x_n \} \subset X$ be a divergent sequence. Is it always true that there exist $\delta > 0$ and a strictly increasing sequence $\{ n_k \} \subset \mathbb{N} $ such that

$$\| x_{n_{k+1}} - x_{n_k} \| > \delta, \quad k = 1, 2, \dots?$$

I can prove that the conclusion is valid in the following situations:

1. $\left\{ x_{n} \right\}$ is unbounded.

Inductive selection $$\left| \left| x_{n_{k+1}} \right| \right| \geqslant \left| \left| x_{n_{k}} \right| \right| +1,k=1,2,\cdots,$$then $$\left| \left| x_{n_{k+1}}-x_{n_{k}} \right| \right| \geqslant 1,k=1,2,\cdots .$$

2. $\left\{ x_{n} \right\}$ has a limit point.

Let $a$ be the limit point, take the neighborhood $U$ of $a$ so that $U^{c}$ contains infinite number of points in $\left\{ x_{n} \right\}$. Just take the points near $a$ and the points outside $U$ in sequence.

So the most difficult situation is that $\left\{ x_{n} \right\}$ is an isolated bounded closed set.