Show that the real vector space of all continuous real valued functions on the interval $[0,1]$ is infinite dimensional.
attempt: Suppose there is a sequence of real valued functions on V, that is $f_1 , f_2 , f_3,....$ such that $f_1,f_2,....,f_m$ is linearly independent $\forall m\in \mathbb{N}$. So then there exists $a_1,...,a_m \in \mathbb{R}$ such that $a_1f_1 +a_2f_2 + ....+ a_mf_m = 0$, I am not sure how to continue . I am trying to show that all the coefficients are zero but I am not sure why and then conclude V is infinite dimensional . Can someone please help me? Thank you!