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Is this true?

Bounded ⇒ totally bounded

No — it is false.

The counterexample

with the discrete metric

Everything is within distance 1, but no finite set of balls of radius ½ covers it.

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this
compactnessmetric spaces

Others that fail the same way

From Metric Spaces › Compactness: open covers, sequential, Heine–Borel

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