Is the extended real line a metric space? I've got a question reading the demonstration of the Theorem 3.2 in POMA of Rudin. Indeed, he says that every convergent sequence in a metric space is bounded.
My question is:

Is $\bar{\mathbb{R}}$ with the usual distance a metric space? 

Indeed, the sequence $(u_n)_{n \in \mathbb{N}}$ defined by $u_0 = +\infty$ and then $u_n = \frac{1}{n} \forall n \geq 1$ is convergent but not bounded... Thus I guess that it is not.
 A: No, $\overline{\mathbb{R}}$ with the usual distance is not a metric space. It is better understood as a topological space (with the order topology).
Although it is not a metric space with the usual distance, it is metrizable. Think about a way to put a metric on it.
But I will disagree with one of the comments (the one that says that $\overline{\mathbb{R}}$ is weird) and will present some arguments for that:


*

*$\overline{\mathbb{R}}$ is compact. This is a great property. This also provides some more insight into the Bolzano-Weierstrass: EVERY sequence in $\overline{\mathbb{R}}$ has a convergent subsequence. This says that every sequence has either a subsequence converging to a real number, or a subsequence converging to some $\pm \infty$.

*Every non-empty subset of $\overline{\mathbb{R}}$ has a $\sup$.

*$\overline{\mathbb{R}}$ clears up the "converging to $\infty$" context, which is generally badly explained by a first course in real analysis. The definition for "converges for $x \rightarrow \infty$" or "converges to $\infty$" is seemed as artificial, and some people even say that "this is not a true convergence", and will even state: We will say $x_n \rightarrow +\infty$, but the sequence actually does not converge.

*$\limsup$ does not have trivialities in his definition. For example, it is common to define that $\limsup$ is the $\sup$ of the set of numbers which are limits of subsequences of the given sequence. By the first item, this set is always non-empty. And by the second item, we always have the $\sup$. This happens naturally, and one doesn't need to define things artificially.

*It even helps to prove that every continuous bijective function on an interval is an homeomorphism.


In my opinion, $\overline{\mathbb{R}}$ is not only not weird, but actually the right place to study analysis.
I made a blog post some time ago about this very subject.
