Closed set with rationals question

[vancouver_water]

Homework Statement

If A is a closed set that contains every rational number in the closed interval [0,1], show that [0,1] is a subset of A.

Homework Equations

The Attempt at a Solution

I'm confused because for the set A = all rationals in [0,1], every point is a boundary point so the set is closed. but clearly [0,1] is not a subset of A. This question is from Spivaks Calculus on Manifolds, question 1-19

 

[Samy_A]

every point is a boundary point so the set is closed

Though the boundary of a set is closed, that doesn't mean that a set A consisting only of boundary points is necessarily closed (the boundary of A could contain points outside of A too).
The set of all rationals in [0,1] is not closed.

You could try to prove that any (irrational) number in [0,1] is in the closure of the rationals in [0,1].

 

[vancouver_water]

oh right I didn't realize the rationals was not closed, I think I got it now, Thanks!