Space of polynomials is isomorphic to a proper subspace of itself

Question

The following question is from Linear Algebra, Kaye and Wilson:

For each integer $n \ge0$, define $g_n(x)=x^n$

$1)$ Define $B:=\{g_n:n\ge 0\}$ show that B is a basis for $\mathbb{F}[x]$.

$2)$ Define the map $\phi:\sum \lambda_i g_i \to \sum\lambda_i g_{i+1}$ is injective but not surjective.

$3)$ Deduce that $\mathbb{F}[x]$ is isomorphic to a proper subspace of itself.

My struggle

Personally I think $1)$ and $2)$ are rather easy to prove but how would they be helpful to prove $3)$?

Answer 1

Since $\phi$ is injective but not surjective, $\phi\bigl(\mathbb F[x]\bigr)$ is a proper subsepace of $\mathbb F[x]$ which is isomorphic to it (and $\phi$ is an isomorphism between them).

Answer 2

The key fact is that the image of $\phi$ is a subspace of $\Bbb{F}[x]$. The map $\phi$ is surjective onto this subspace, and it is injective, hence it is an isomorphism.