Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 12E / a / ii

No. Give an infinite set $X$ the discrete metricThis space is bounded. Every Cauchy sequence is eventually constant, so it is complete. However, the open cover by the singleton balls $B_{1/2}(x)=\{x\}$ has no finite subcover. Hence $X$ is not compact.