linear algebra - 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 $a_k \in \mathbb R$