Page 30 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Definition 1: Interval
An interval in R is an non-empty subset of R which has the property that, whenever two numbers
a and b belong to it, all numbers between a and b also belong to it.

The set N of natural numbers is not an interval because while 1 and 2 belong to N, but 1.5 which
lies between 1 and 2, does not belong to N.
We now discuss various forms of an interval.

Let a, b  R with a  b.
(i) Consider the set {x  R : a  x  b}. This set is denoted by ]a, b[, and is called a closed
interval. Note that the-end points a and b are included in it.

(ii) Consider the set{x  R : a < x < b}. This set is denoted by [a, b], and is called an open
interval. In this case the end points a and b are not included in it,
(iii) The interval {x  R: a  x < b} is denoted by [a, b[.

(iv) The interval {x  R : a < x  b} is denoted by ]a, b].
You can see the graph of all the four intervals in the Figure 2.1.

Figure 2.1

Intervals of these types are called bounded intervals. Some authors also call them finite
intervals. But remember that these are not finite sets. In fact these are infinite sets except for the
case [a, a] = {a}.
You can easily verify that an open interval ]a, b[ as well as ]a, b] and [a, b[ are always contained
in the closed interval [a, b].

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