Compact set in $(\mathbb R,\rho_1)$

Question

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by Bolzano–Weierstrass theorem, if sequence is bound, you can always choose convergent sequence.

Answer 1

Let's be more precise:

by Bolzano–Weierstrass theorem, if sequence is bounded, you can always choose convergent sequence in the Euclidean metric.

Therein lies the problem, which Henno Brandsma warned you about. The concept of convergence depends on the metric we use. Being convergent in Euclidean metric does not necessarily imply convergence in the $\rho$ metric.

What does convergence in $\rho$ metric mean? It means $\rho(x_n,x)\to 0$ where $x$ is the limit of the sequence. Looking at the formula for $\rho$, we see that this can happen only if either $x_n=x$ or both $x_n$ and $x$ are small. This makes it really hard to converge to something that is not $0$.

Suggestion: Consider the sequence $x_n= 1-\dfrac1n$, $n=1,2,3,\dots$ It converges to $1$ in the Euclidean metric, but what happens in the $\rho $ metric?

Further hint: show that for any two distinct indices $n,m\ge 2$ we have $\rho(x_n,x_m) \ge 1$.