Module 1
:
Real Numbers, Functions and Sequences
Lecture 1
:
Real Numbers, Functions [ Section 1.1 : Real Numbers ]
Practice Exercises 1.1 : Real Numbers
(1)
Using the Principle of Mathematical Induction, prove the following for all n
:
(i)
n > 1
implies
n - 1
.
(ii)
For
x
with
x > 0
, if
x + n
, then
x
.
(iii)
m + n, m n
for all m,n
.
(2)
Using the Principle of induction prove the following:
(i)
If
then
and
.
(ii)
If
and
, then
for all
.
(3)
Show that the product of any two even (odd) integers is also an even (odd) integer.
(4)
Find an irrational number between 3 and 4.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
then 
and 
. 
and 
, then
for all 
. 
