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Examples

For example, to prove that a continuous function is bounded, you either find a neighbourhood of each point on which by finitely many such neighbourhoods (by the HeineBorel theorem), or you assume that, apply the BolzanoWeierstrass theorem, and reach a contradiction.

If you want to prove something using the compactness of, then you can either prove it directly using the HeineBorel theorem or you can prove it by contradiction by reformulating everything in terms of sequences and applying the BolzanoWeierstrass theorem.

From HeineBorel we know that a continuous function on a closed interval takes a global maximum and a minimum.

1984, “HeineBorel does not imply the Fan Theorem”,
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