Dimension of vector space of real-valued functions over $R$ I'm trying to prove that the space of real-valued functions on the closed interval $[a,b]$ where $a < b$ is an infinite dimensional vector space over $\mathbb{R}$. 

$V$: The space of real-valued functions on the closed interval.
$W$: Subset of $V$ where all the coefficients are set to 1.
If I consider the polynomials $1,x,x^2,~...~x^n$ (Where all these polynomials are linearly independent), I can repeatedly create a polynomial $x^{n+1}$ which is linearly independent, and thus the basis is of infinite order for $W$, $W$ is infinite dimensional and this implies that $V$ is infinite dimensional. 
Am I getting anywhere? Would appreciate feedback. 

This does not make sense to talk about a basis for $W$ since $W$ is not a subspace (not stable by sum). You don't need to introduce anything new, simply to show that $x^{n+1}$ is not in the span of $\{1,x, \dots, x^n\}$, this will shows that a basis of $V$ can't be finite. 
