Consider a real sequence $x\_n$ and the set $A = \{ r | \liminf x\_n < r < \limsup x\_n \, , \, r \in \mathbb Q\}$. Is it true that $|A| = 0$ if and only if $x\_n$ converges to a real limit?

My attempt:

Note that if $\liminf x\_n < r < \limsup x\_n$ for some real $r$ then there exists a $q \in \mathbb Q$ close enough to $r$ such that $\liminf x\_n < q < \limsup x\_n$ since the rationals are dense in the reals. Also, if there exists a rational $q$ such that $\liminf x\_n < q < \limsup x\_n$ then the inequality is satisfied for a real value since $\mathbb Q \subset \mathbb R$.

So, the set $A$ above has cardinality $0$ if and only if the set $B$ has cardinality $0$ where $B = \{r | \liminf x\_n < r < \limsup x\_n, r \in \mathbb R\}$.

But $B$ has cardinality $0$ only when $\liminf x\_n = \limsup x\_n$ which implies that $x\_n$ converges to a real limit.

I could use some feedback on my reasoning.