Hello student,

A subset S of a metric space (M, d) is bounded if there exists r > 0 such that for all s and t in S, we have d(s, t) < r. The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness.

Moreover, For example, a bounded subset of the real line is totally bounded. On the other hand, if ρ is the discrete metric on an infinite set X, then X is bounded but not totally bounded.

Good luck...