Even Natural Numbers are Infinite
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Theorem
The set of even natural numbers is infinite.
Proof
Let $E$ denote the set of even natural numbers.
Aiming for a contradiction, suppose $E$ is finite.
Then there exists $n \in \N$ such that $E$ has $n$ elements.
Let $m$ be the greatest element of $E$.
But then $m + 2$ is an even natural number.
But $m + 2 > m$, and $m$ is the greatest element of $E$.
Therefore $m + 2$ is an even natural number that is not an element of $E$.
So $E$ does not contain all the even natural numbers.
From that contradiction it follows by Proof by Contradiction that $E$ is not finite.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity?