Real Analysis Questionnaire

Description

1. Let (xn)1

n=1 be a sequence of real numbers. Express the following two statements using quantifiers:

(a) The sequence (xn)1

n=1 is non decreasing.

(b) The sequence (xn)1

n=1 is bounded from above.

Show that if the sequence (xn)1

n=1 satisfies both these properties, then limn!1 xnexists and is equal to

supn2N xn.

2. For any two vectors in Rn, x = (x1; : : : ; xn), y = (y1; : : : ; yn), we define

d1(x; y) := max

i=1;:::;n

jxi 􀀀 yij:

(a) show that d1 is a metric on Rn.

(b) Show that a sequence (x(k))1

k=1 converges to x 2 Rn with respect to d1 if and only if for every

i = 1; : : : ; n , the sequence (x(k)

i )1

k=1 converges to xi (in R with respect to the usual absolute value

metric).

3. For a metric space (X; d), the diameter of a subset A X is defined by

(A) := sup

x;y2A

d(x; y):

Here we allow (A) = 1. Show that if A;B X satisfy A \ B ̸= ∅, then (A [ B) (A) + (B). Give

an example to show this is not necessarily true when A \ B = ∅.

Exercises from the book.

1.1.5, 1.1.6, 1.1.13, 1.1.14, 1.1.16