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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 Heine-Borel theorem), or you assume that, apply the Bolzano-Weierstrass theorem, and reach a contradiction.
If you want to prove something using the compactness of, then you can either prove it directly using the Heine-Borel theorem or you can prove it by contradiction by reformulating everything in terms of sequences and applying the Bolzano-Weierstrass theorem.
From Heine-Borel we know that a continuous function on a closed interval takes a global maximum and a minimum.
1984, “Heine-Borel does not imply the Fan Theorem”,