Problem
Prove that the real vector space of all continous real-valued functions on the interval $[0,1]$ is infinite-dimensional
Solution
Construct the linearly independent sequence of functions in $\mathbb{R}^{[0,1]}$ $1,z,z^2,z^3, \dots$ where the $j$‘th vector in the sequence is the function $z^{j-1}$.
The existence of such a sequence then implies, by P2A14, that $\mathbb{R}^{[0,1]}$ is infinite dimensional.