Completeness of the continuous-map space in the uniform metric ($C(X,Y)\text{ is complete when }Y\text{ is complete and bounded}$)

A Cauchy sequence in the uniform metric has a pointwise limit in the complete target, and the common Cauchy bound makes convergence uniform. The triangle inequality with one fixed continuous approximant shows that the limit is continuous. Thus the space of all continuous maps into a bounded complete metric space is complete, with no compactness assumption on the domain.