Proof | Integers are infinitely many

Sunil Syal — Tue, 22 Sep 2020 13:42:31 +0000

Proof - Integers are Infinitely Many

Theorem: Prove that there are infinitely many integers.

Proof: Let us assume that there are finitely many Integers.

\( \begin{aligned} &\Rightarrow \text{∃ a greatest integer, say } x \end{aligned}\)

Let us get a new number say y by adding 1 to x.

\( \begin{aligned} y&=x+1\end{aligned}\)

\( \begin{aligned} &Now \ \ \mathbf{y} \in\mathbb{Z} \\ &\text{because } \mathbf{x+1} \text{ is also an integer}\\\\&But \ \ y > x \end{aligned}\)

So we got another integer y > x.

This is a contradiction to our assumption that x is the greatest integer.

This means our assumption is wrong that there are finitely many integers.

Hence we can conclude that there are infinitely many integers.