Proof function is increasing and express in different form
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Real Analysis - Prove a metric...
Let d be a metric in X. Prove that p(x,y)=(d(x,y))/(1+d(x,y)) is also a metric in X.
Let (X,d) be a metric space. Define a closed ball with center x and radius r to be the set B[x;r]={y:d(x,y)<=r}. Prove that B[x,r] is a closed set.
Real analysis - discrete space both open and closed
Show that every subset of a discrete space is both open and closed.
Let X be the set of all bounded sequences of real numbers. If x=(a_k) and y=(b_k) let d be the metric funtion defined by d(x,y)=sup{|a_k - b_k|} (note _ denotes subscript) Show that the metric space defined above is complete.
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}.
