# Proof: if A is subset of B then closure of A is subset of closure of B

I'm thinking proof by contradiction but I can't seem to get anywhere.

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pwsnafu
Think about limit points. x is in the closure of A, so it is a limit point of A...

Fredrik
Staff Emeritus
Gold Member
You should probably state the definition of "closure" that you would like to use, and maybe also your definition of "closed". I like the definition that says that the closure of E is the smallest closed set that contains E. If we use that definition, the proof is very simple.

I see that you're new here. If you're wondering why we don't just tell you the complete solution, it's because the forum rules tell us not to do that when someone posts a textbook-style question. We are only allowed to give you hints, and tell you if you're doing something wrong.

disregardthat
It might be easier to prove that if A is contained in a closed set (here the closure of B, which follows since B is contained in the closure of B and A is contained in B), then the closure of A is contained in the same closed set.

PAllen
2019 Award
May I ask why this thread is here? It isn't number theory, and it looks like homework to me.

HallsofIvy