Show that (0,1) is homeomorphic to (a,b)

Show that (0,1) is homeomorphic to (a,b)

Real Analysis - bounded open ball

Show that a set E in the metric space X is bounded if and only if, for some "a" in X, there exists an open ball B(a;r) such that E is a subset of B(a;r).

Real Analysis - finite union compact

Show that the finite union of compact sets in a metric space X is compact.

Real Analysis - finite subsets compact

Prove that every finite subset of a metric space is compact.

Real Analysis - continuous function on compact space

Show that if f is a continuous real-valued function on the compact space X, then there exist points x_1, x_2 in X such that f(x_1)=inf{f(x):x in X} and f(x_2)=sup{f(x):x in X}.