Let V be the vector space of functions $f : N → R$. Prove that V is infinite dimensional.
My thoughts: A finite dimensional vector space over a countable field is necessarily countable: if $v_1$,…,$v_n$ is a basis, then every vector in $V$ can be written uniquely as $α_1$$v_1$+⋯+$α_n$$v_n$ for some scalars $α_1$,…,$α_n$$∈f$, so the cardinality of the set of all vectors is exactly $|$f$|^n$. But this is where I'm stuck. I am still unclear on how I can prove it is infinite dimensional.