Oh dear, sorry I didn't the see the pre-university part, please ignore this post! I'll post it in the university section now.
EDIT: I've posted this in the wrong part of the forum, please ignore this question.
Hi, my question is:
Show that the unit sphere with center 0 in R3, namely the set S2 :={x in R3 : (x1)^2+(x2)^2+(x3)^2=1} is closed in R3.
What I've done so far is say that to prove it's closed, I need to show that the complement of S2, X`S2, is open, so:
X`S2 :={x in R3 : (x1)^2+(x2)^2+(x3)^2 is not equal to 1}. I then stated the definition of an open set:
X`S2 is open if for all x in S2 there exists r>0 such that B(x, r) is a subset of S2, B(x, r)={y : d(x, y)<r}.
What I'm stuck on in this question is the choice of r, and how to show that, for any y in B(x,r), y is in S2.
Any help would be appreciated