Where the dense property applied in this proof using compactness?

Question

There exists $t_1, t_2, \ldots, t_k \in \mathbb{Q}\cap[a,b]$ such that $\forall x \in [a,b],\ |x-t_j|<\delta$, for at least one $j = 1,\ldots, k$

My professor provide me with the hint that $\{(t-\delta, t+\delta)\mid t\in \mathbb{Q}\}$ is an open cover and that $\mathbb Q$ is dense.

My idea is: $x$ must be contained in some $(t_i-\delta, t_i+\delta)$ therefore there must be some $t_j$ statisfies the property.

Where did the dense property apply?

Answer 1

The denseness of $\mathbb{Q}$ is what guarantees that $\{(t-\delta,t+\delta):t\in\mathbb{Q}\}$ is actually a cover of $[a,b]$.