Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Show that if a set $A$ in a metric space is bounded, so is each subset $B \subseteq A$.

share|cite|improve this question
What does it mean for $A$ to be bounded? I.e., how would you write that out explicitly? – Jonas Meyer Feb 27 '11 at 5:26

If $A$ is bounded then there exists $x \in M$ and $r>0$ such that for all $a \in A$, we have $d(x,a) <r$. So how would you show that each subset $B \subseteq A$ is bounded?

share|cite|improve this answer

Definition: A metric space $M$ is called bounded if there exists some number $r>0$, such that $d(x,y) \leq r$ for all $x, y \in M$.

Since you are assuming $A$ to be bounded then there is some number $k$ such that $d(x,a) \leq k$ for all $x,a\in A$. Since $B \subset A$ and this condition happens for all elements in $A$, therefore it should happen for elements in $B$ as well.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.