Page 58 - DMTH401_REAL ANALYSIS

Real Analysis

Notes This concludes the proof of the property.
You can try the following exercise similar to this property.
Now let us see another interesting relationship between the inequalities and the modulus of a
real number.
By definition, |x| is a non-negative real number for any xR. Therefore, there always exists a
non-negative real number u such that
either (x| < u or |x| > u or |x| = u.
Suppose |x| < u. Let us choose u = 2. Then

|x| < C  Max. {-x, x} < 2
–x < 2, x < 2

x > –2, x < 2
–2 < x, x < 2
–2 < x <2.

i.e. |x| < 2 –2 < x < 2
Conversely, we have

–2 < x < 2 –2 < x < 2
2 > –x, x < 2
–x < 2, x < 2

Max. {–x, x} < 2
|x| < 2.
i.e.

–2 < x < |x| < 2
Thus, we have shown that
|x| <  –2 < x < 2.

This can be generalised as the following property.
Property 5: Let x and u be any two real numbers.

|x|  u  –u  x  u.
Proof: |x|  u  Max. {–x, x}  u
 –x  u, x  u

 x  –u, x  u
 –u  x, x  u

 –u  x  u
which proves the desired property.
The property 5 can be generalized in the form of the following exercise.

52 LOVELY PROFESSIONAL UNIVERSITY