Proving $|X^{\mathbb{R}}| \lt |\mathbb{R}^X|$ for $X=\mathbb{R}^{\mathbb{R}}$

Question

Let $X=\mathbb{R}^{\mathbb{R}}$.

Claim: $|X^{\mathbb{R}}| \lt |\mathbb{R}^X|$

How can this be shown? Note: We are assuming AC and $A^B$ represents the set of functions from B into A

Answer 1

You don't need the axiom of choice for this.

HINT: First, recall that $|A^B|=|A|^{|B|}$ and that $(X^Y)^Z=X^{YZ}$ in cardinal arithmetic. Now, note that $2^X\leq\Bbb R^X$, but that $|X^\Bbb R|=|X|$.

(This is the same proof as the proof that $\Bbb{|R^N|<|N^R|}$.)