Background: I'm a computer science major, but who has done a lot of math (real analysis, linear/abstract algebra, combinatorics, probab&stats, numerical analysis, linear programming) and currently doing undergraduate research in computational algebra/geometry.
I'm taking a graduate level...
Awesome explanation, thank you. I knew it was something about bijections, but I couldn't think of any example for the second inequality for some reason.
From page 89 of baby Rudin:
"Theorem Suppose f is a continuous mapping of a compact metric space X into a metric space Y. Then f(x) is compact.
(truncated)
Note: We have used the relation f(f^{-1}(E))\subsetE, valid for E \subset Y. If E\subsetX, then f^{-1}(f(E))\supsetE; equality need...
on page 48 of baby Rudin, it says " the sequence {1/n} converges in R1 to 0, but fails to converge in the set of all positive real numbers [with d(x,y) = |x-y|]."
ok, I know it has something to do with 1/n going to infinity near zero, but it does that whether the metric space is R1 or just...