Cardinality of $ X\times \mathbb{N} $ when $X$ is infinite

Question

Suppose $X$ is infinite. I think we can show that $|X\times\mathbb{N}| = |X|$. How do we construct an injective function $f:X\times\mathbb{N} \to X$?

Answer 1

When $X$ is infinite you can partition as $X=\sqcup_{i\in I}X_i$, each $X_i$ being in one-to-one correspondence with $\mathbb{N}$. Let us call $\phi_i : X_i\rightarrow \mathbb{N}$ such a bijection. Then, the "trick" is to construct a bijection between $X_i\times \mathbb{N}$ and $X_i$ which can be done through any bijection between $\mathbb{N}\times \mathbb{N}$ and $\mathbb{N}$.