posted 11 months ago

posted 11 months ago

\text { (a) Let } d: X \times X \rightarrow \mathbf{R}_{+} \text {be defined as }

d(x, y)=\left\{\begin{array}{ll} 0 & \text { if } x=y \\ 1 & \text { if } x \neq y \end{array}\right.

Show that d is a metric on X. This metric is called discrete.

(b) Let d be a metric on X and let f : R4 → R4. If ƒ is strictly increasing, is it true that fod is a metric on X?

posted 11 months ago

posted 11 months ago

(a) Let {rn} be a converging sequence in a metric space X and let x € X be its limit. Use the definition of compactness to show that the set {x}U{xn} is compact.

(b) Show that a subset of a metric space X is closed if and only if its intersection with every compact subset of X is closed.

posted 1 years ago

posted 1 years ago

posted 1 years ago

s_{n+1}=\sqrt{2+\sqrt{s_{n}}} \quad(n=1,2,3, \ldots)

prove that {s.} converges, and that s. <2 for n=1, 2, 3, ....%3D

posted 1 years ago

posted 1 years ago

posted 1 years ago