Polynomial equal to polynomial of lower degree I am studying Linear Algebra Done Right, chapter 2 problem 6 states:

Prove that the real vector space consisting of all continuous real valued functions on the interval $[0,1]$ is infinite dimensional.

My solution:
Consider the sequence of functions $x, x^2, x^3, \dots$ 
This is a linearly independent infinite sequence of functions so clearly this space cannot have a finite basis.
However this prove relies on the fact that no $x^n$ is a linear combination of the previous terms. In other words, is it possible for a polynomial of degree $n$ to be equal to a polynomial of degree less than $n$. I believe this is not possible, but does anyone know how to prove this? More specifically, could the following equation ever be true for all $x$?
$x^n = \sum\limits_{k=1}^{n-1} a_kx^k$ where each $a_k \in \mathbb R$

Suppose $x^n = \sum\limits_{k=1}^{n-1} a_kx^k$. Consider what happens when $x$ is much larger than  $|a_1|+\ldots+|a_{n-1}|$. By the triangle inequality, you will get a contradiction. Essentially, you are showing that $x^n$ grows faster than any degree $n-1$ polynomial...
