Geodesic
On Closed, Totally Bounded Sets
Canadian mathematical bulletin, Tome 9 (1966) no. 4, pp. 525-526

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C. Goffman asserts that "... in a metric space X a set S is compact if and only if it is closed and totally bounded." [1] and "... every totally bounded sequence in a metric space has convergent subsequence." [2].The statements (incidentally, equivalent to each other) are both wrong, as the following counter-example shows. Take the set of all reals in the open interval (0, 1) with the usual metric. This space is closed and totally bounded, but not compact. 
Murdeshwar, M. G. On Closed, Totally Bounded Sets. Canadian mathematical bulletin, Tome 9 (1966) no. 4, pp. 525-526. doi: 10.4153/CMB-1966-066-1
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     title = {On {Closed,} {Totally} {Bounded} {Sets}},
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