3 20 points Mark each of the following TRUE or FALSE If a st

3. (20 points) Mark each of the following TRUE or FALSE. If a statement is false, give an example that shows it is false. (a) Every sequentially compact set is bounded (b) Every continuous function is bounded (c) Suppose that f : D R is uniformly continuous. Then f is continuous. (d) Suppose that f : D R is continuous. Then f is uniformly continuous (e) Every sequence in a sequentially compact set converges. (f) Every sequence in a closed set converges. (g) Every closed set is bounded. (h) Let K be a sequentially compact set of real numbers. Let f : K R be continuous. Then f is bounded. (i) Let K be a sequentially compact set of real numbers. Let f:K R be continuous. Then f is uniformly continuous. (j) Suppose that f : [0, oo)-R is continuous and f(x) 0, Then f(0)3.

Solution

a)TRUE

b)FALSE - It is true only when there closed bound on interval.For example, f:[0,inf)->R =x,here x is continuous but not bounded.

c)True

d) False,Ex f(x)=x² in [0,inf) is conitnuous but not uniformly continuous

e)True

f)False, Every sequence in a closed and bounded set is converges.Ex f(x)=tan(x),is closed on set(0,pi/2) but diverges.

g) False

Reason - The Z i.e integer are the subset of Real numbers R but it is not bounded.Ex U x:z,[2x,2x+1]

h) True

i)True

j) True