22
Sep
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.
