Neighborhoods in 𝐑

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The Real Number Line

One way to represent the real numbers is on the real number line as depicted below.


Real number line


We will now state the important geometric representation of the absolute value with respect to the real number line.

Definition: If and are real numbers, then we say that the distance from to the origin is the absolute value of , . We say that the distance between and is the absolute value of their difference, namely .

For example consider the numbers and . There is a distance of in between these numbers because .

Epsilon Neighbourhood of a Real Number

Definition: Let be a real number and let . The -neighbourhood of the number is the set denoted . Alternatively we can define .


-neighbourhood around () expressed on the real number line


For example, consider the point , and let . Then .

We will now look at a simple theorem regarding the epsilon-neighbourhood of a real number.

Theorem 1: Let be a real number. If , then .
  • Proof of Theorem 1: Suppose that for some , , . We know that then if and only if and therefore .

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