Homework Statement
show that x_n converges to x if and only d(x_n, x) converges to 0.
Homework Equations
|x_n - x| < ε for all ε>0
The Attempt at a Solution
well d(x_n,x) converges to 0 if d(x_n,x)<ε
i just don't know how to relate that back to |x_n - x|
So I guess what I'm really wondering is can a set be unbounded if it doesn't go to infinity? That's what I can't seem to wrap my brain around, because I feel that if a set doesn't go to infinity, there will always be a real number K larger than the members of the set.
Is a bounded set synonymous to a set that goes to infinity? I feel like unless a set is
(-infinity, n) or [n, infinity) it is not going to be unbounded.
The other thing that I was wondering is can a set be neither open nor closed AND unbounded? Doesn't the definition of open/closed imply...