Prove: $|\mathbb{N}|\leq |\mathbb{R}|$

Question

Prove: $|\mathbb{N}|\leq |\mathbb{R}|$

Note: $|\cdot|$ denotes cardinality.

My work

Suppose $|\mathbb{N}|\geq |\mathbb{R}|$

We can write $\mathbb{R}=\mathbb N\cup(\mathbb Q-\mathbb N)\cup\mathbb{I}$ then $$|\mathbb{R}|=|\mathbb N\cup(\mathbb Q-\mathbb N) \cup \mathbb{I}| = |\mathbb{N}| + |\mathbb{Q-N}| + |\mathbb{I}| = |\mathbb{N}| + |\mathbb{Q}| - |\mathbb{N}|+|\mathbb{I}|$$

Then,

$$|\mathbb{N}|+|\mathbb{Q}|-|\mathbb{N}|+|\mathbb{I}|\leq|\mathbb{N}|\implies |\mathbb{Q}|+|\mathbb{I}|\leq|\mathbb{N}|$$

Here i'm stuck. Can someone help me?

Answer 1

The embedding map $j:\mathbb{N}\to\mathbb{R}$ given by $j(x)=x$ is trivially an injection and the claim follows.

Answer 2

Since $\mathbb{N}\subset\mathbb{R}$, $|\mathbb{N}|\leqslant|\mathbb{R}|$.