TITLE: Disproving that a sequence in $\mathbb{R}^{n}$ converges
QUESTION [1 upvotes]: Let $\{x_{n}\}_{n\in \mathbb{N}}$ a bounded sequence in $\mathbb{R}^n$. Let assume that the sequence $\{d(x_n,0)\}_{n\in \mathbb{N}}$ (In the euclidean space) is strictly increasing.
Prove/Disprove: $\{x_{n}\}_{n\in \mathbb{N}}$ converges

I am trying to grasp it conceptually but did not manage.
The sequence $\{x_{n}\}_{n\in \mathbb{N}}$ are bounded, therefore each element of the sequence is bounded and we can look at it, as a sequence of points in a closed ball in $\mathbb{R}^n$.
Now we take the distance of the a point from the origin, and the distance is strictly increasing, but how can it be if the points themselves are bounded?

REPLY [1 votes]: Consider the following sequence in $\mathbb{R^d}$: for $n\geq 1$,
$$x_n=\left((-1)^n\left(1-\frac{1}{n}\right),0,\dots,0\right).$$
Does it satisfy the hypothesis?
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