Page 59 - DMTH401_REAL ANALYSIS

Unit 3: Matric Spaces

Notes

Task For any real numbers x, a and d,

|x – a|  d  a – d  x  a + d.

Example: Write the inequality 3 < x < 5 in the modulus form.
Solution: Suppose that there exists real numbers a and b such that

a – b = 3, a + b = 5.
Solving these equations for a and b, we get

a = 4 > b = 1.
Accordingly,
3 < x < 5  4 – 1 < x < 4 + 1

 –l < x – 4 < l

|x – 4| < 1

Task 1. Write the inequality 2 < x < 7 in the modulus form.

2. Convert |x – 2| < 3 into the corresponding inequality.

3.3 Neighbourhoods

You are quite familiar with the word 'neighbourhood'. You use this word frequently in your
daily life. Loosely speaking, a neighbourhood of a given point c on the real line is a set of all
those points which are close to c. This is the notion which needs a precise meaning. The term
'close to' is subjective and therefore must be quantified. We should clearly say how much 'close
to'. To elaborate this, let us first discuss the notion of a neighbourhood of a point with respect to
a (small) positive real number .

Let c be any point on the real line and let 6 > 0 be a real number. A set consisting of all those
points on the real line which are at a distance of 6 from c is called a neighbourhood of c. This set
is given by
{x R : |x – c| < 6}

= {x R : c –  < x < c + 6)
= ]c – 6, c + [

Which is an open interval. Since this set depends upon the choice of the positive real number ,
we call it a 6-neighbourhood of the point c.

Thus, a -neighboured of a point c on the real line is an open interval ]c – 6, c + 6[,  > 0 while c
is the mid point of this neighbourhood. We can give the general definition of neighbourhood of
a point in the following way.

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