I want to prove that the linear space of all continuous functions from $ \mathbb{R} $ to $ \mathbb{R} $ is not finite-dimensional. Can you verify my proof?
Define $ A = \{f \in \mathbb{R}^\mathbb{R} \mid f \text{ is continuous} \}, V = (A, +, \cdot) $, and suppose $ A $ is finite-dimensional. Therefore, there exists $ n \in \mathbb{N} $ such that $ dimV = n \Rightarrow \forall B \subseteq A(\left|B\right| > n \Rightarrow B \text{ is lin. dependent})$. Choose $ B = \{1, x, x^2, x^3, ...\}$ which is lin. independent. However, $ |B| > n $ - contradiction.
You could maybe also show why exactly $\{1,x,x^2,...,\}$ are linearly independent, but the proof works.