Mathematics Homework Help
Mathematics Homework Help. 8 real analysis textbook question
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