Enumeration of an Infinite Countable Set

Let $\mathbb{N}$ be the set of natural numbers, with the order of that definition. A sequence in $\mathbb{N}$ is strictly increasing as defined there. The notions finite, countable and bijection are those of the indicated definitions.

1. (Subsets of the natural numbers)¶ Let $A\subseteq\mathbb{N}$ be a set that is not finite. Then there is a strictly increasing sequence $(t_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ such that the map $k\mapsto t_{k}$ is a bijection from $\mathbb{N}$ onto $A$.

2. (Countable sets)¶ Let $X$ be a countable set that is not finite. Then there is a bijection from $\mathbb{N}$ onto $X$.