Prove that $\displaystyle |[0,1]| < |\mathbb{R}^{[0,1]}|$.

Attempted Solution:

To prove this inequality I need to show there exists a 1-1 function $\displaystyle f: [0,1] \to \mathbb{R}^{[0,1]}$. And that all functions $\displaystyle f: [0,1] \to \mathbb{R}^{[0,1]} $ cannot be ONTO.

I'm having trouble conceptualizing a 1-1 function that maps from $\displaystyle [0,1] \to \text{set of all functions: } g:[0,1] \to \mathbb{R} $. However to the best of my knowledge I need a function that takes values from the interval [0,1] as input and outputs a function from the set of functions $\displaystyle \mathbb{R}^{[0,1]} $. A simple function in $\displaystyle \mathbb{R}^{[0,1]} $ that I'm thinking of is just a straight line, so $\displaystyle g(x) = x + b$ where b acts like the y-intercept. So my idea for a 1-1 mapping from $\displaystyle f: [0,1] \to \mathbb{R}^{[0,1]}$, would take inputs from [0,1] and map them to various parallel lines, $\displaystyle g(x)$ with different values for b. So basically $\displaystyle f(b) = x + b, \text{ where }x, b \in [0,1]$. So my function is mapping to various parallel lines, I think that insures that f is 1-1.

If my 1-1 function is indeed correct, now I need to prove all functions $\displaystyle f: [0,1] \to \mathbb{R}^{[0,1]} $ cannot be ONTO. And I can't think of a way to start making progress on this part.

Thanks in advance for the help.