Why to define compactness with sequence?

Question

We defined compactness by can be covered by finite open sets, then we introduced a sequence definition.

Why to define compactness again with sequence? What's the use of sequentially compactness?

Answer 1

You can define compactness for sets contained in more general spaces than just $\mathbb{R}$. And then, there is a difference between compactness and sequential compactness.

Answer 2

When applicable, sequential compactness is often much easier to prove than compactness.