Sequentially compactness of $\mathbb{Q}\cap [a,b]$

Question

Let $a\leq b$ be real numbers. Can the set $\mathbb{Q}\cap [a,b]$ be sequentially compact? It is possible for me to to prove that $\mathbb{Q}\cap [0,1]$ is not sequentially compact.

The metric is the Euclidean metric on $\mathbb{R}$.

Answer 1

It's clear that $\mathbb{Q}\cap [a,b]$ is sequentially compact if $a=b\in\mathbb{Q}$. Note that $\mathbb{Q}\cap [a,b]$ is empty when $a=b\notin\mathbb{Q}$.

If $a<b$, there exists an irrational number $q$ such that $a<q<b$ by dense of irrationals and then we can construct a sequence of rationals on $[a,b]$ that converges to $q$. Hence $\mathbb{Q}\cap [a,b]$ is not sequentially compact.