TITLE: Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional.
QUESTION [17 upvotes]: Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional.
Clearly it's infinite dimensional, because if you consider say $P (\mathbb{F})$ on $[0,1]$, then there are an infinite amount of continuous real-valued functions on the interval, but how do I prove this?

REPLY [5 votes]: The plethora of answers already submitted are more than sufficient to answer the question. Interestingly, one may actually show that the collection of smooth functions on any smooth manifold forms an infinite dimensional vector space. Since $C^{\infty}(M) \subseteq C(M)$ this would also prove your result.
To prove that $C^\infty(M)$ is infinite dimensional requires a bit of work, but is actually quite intuitive. To show that a vector space $V$ is infinite dimensional, it is sufficient to show that for any $k \in \mathbb N$, one may find $k$-linearly independent vectors. Fix an arbitrary chart $\phi: U \subseteq M \to \mathbb R^n$ and take $k$-distinct points $\{x_i\}_{i=1}^k$ in $\phi(U)$. By Hausdorffness of $\mathbb R^n$, one may find open neighbourhoods $X_i$ of each $x_i$ and consequently $\epsilon_i$ such that $B_i:=\overline{B_{\epsilon_i}(x_i)} \subseteq X_i$. If $f_i:\mathbb R^n \to \mathbb R$ are bump functions on each $B_i$ and $W_i = \phi^{-1}(B_i)$, define the continuous functions
$$ g_i: M \to \mathbb R, \qquad g_i(x) = \begin{cases} f_i(\phi(x)) & x \in W_i \\ 0 & x \notin W_i \end{cases}.$$
These may be extended to smooth functions $\hat g_i$ on $M$ such that $g_i|_{W_i} = \hat g_i|_{W_i}$, and these $\hat g_i$ (of which there are $k$) are easily seen to be a linearly independent set in $C^\infty(M)$.