# Thread: Metric Spaces

1. ## Metric Spaces

Need some help.

1. Prove that every finite subset of a metric space, X, is closed.
On this, am I trying to prove that X is complete by showing that every Cauchy sequence converges to some point in X?

2. How do I prove that Q is not closed in R?

Thanks.

2. Originally Posted by taypez
1. Prove that every finite subset of a metric space, X, is closed.
2. How do I prove that Q is not closed in R?
“On this, am I trying to prove that X is complete by showing that every Cauchy sequence converges to some point in X?” has nothing to do with #1.
If F is a finite subset of a metric space X just show that the complement of F is an open set. For any point t not in F it is easy to construct an ball centered at t containing no other point of F.

For #2, you know that every irrational number is the limit of a sequence of rational numbers. Put another way: Every irrational number is a limit point of the set of rational numbers.