Real Analysis : Open and closed discrete spaces

Show that every subset of a discrete space is both open and closed.

Real Analysis : Bounded Sequences, Metrics and Completeness

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 Balls

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}.

Real Analysis : Compact and Complete Metric Space

If X is compact prove that C(X,R) is a complete metric space.

Real Analysis : In this space, is every closed and bounded set compact?

Let (X,d) be the metric space consisting of m-tuples of real numbers with metric d(x,y)=max{|a_k-b_k|:k=1...m} where x={a_1, a_2,...,a_m} and y={b_1, b_2,...,b_m}. In this space is every closed and bounded set compact? keywords: Heine–Borel, Borel

Real Analysis - Show E is equicontinuous.

Let E be a set of differentiable functions in C[a,b] with uniformly bounded derivatives; i.e., there exists a number M, independent of f in E, such that |f'(x)|<=M for all x in [a,b] and all f in E. Show that E is equicontinuous.