Infinite Set is Equivalent to Proper Subset/Examples/Even Integers

Examples of Use of Infinite Set is Equivalent to Proper Subset
Let $\Z$ be the set of integers.

By Integers are Countably Infinite, $\Z$ is infinite.

Let $E$ be the set of all even integers.

We have that, for example, $3 \in \Z$ but $3 \notin E$

Hence $E$ is a proper subset of $\Z$

Let $f: \Z \to E$ be the mapping defined as:
 * $\forall x \in \Z: \map f x = 2 x$

Then $f$ is a bijection.

Hence $\Z$ and $E$ are equivalent.