is bounded, so if compactness were only about boundedness it would qualify. Show directly that it is not compact by exhibiting an open cover with no finite subcover.
Interactive Explainer
Why (0,1) is not compact — an explicit cover with no finite subcover
Metric SpacesAnalysis & Linear Algebrafree
Not an appeal to Heine–Borel: the actual cover, and why every finite subfamily misses points.