$C^0([a,b])$ is an infinite dimensional vector space I am proving that $C^0([a,b])$ is an infinite dimensional vector space. The fact that it is a vector space is clear. But I cannot understand how to prove that it has infinite dimension.
Let $F:=\{f_n:[a,b]\to\mathbb{R},x\mapsto x^n:n\in\mathbb{N}\}$. $\mid F\mid=+\infty$. But how can I prove that $F$ is made of linearly independent vectors?

If some finite linear combination
$$\sum_{k=0}^n a_kx^k$$
is the function $0$, then the polynomial
$$\sum_{k=0}^n a_kx^k$$
has infinitely many roots, namely, the elements of $[a,b]$. That means that the polynomial is $0$, that is, $a_k=0$ for any natural $k$.
Hence, $F$ is lineary independent.
