Kernel and Image of a linear transformation $L:P_n \rightarrow P_n$ Given a linear transformation
$L: P_n \rightarrow P_n$, where $(Lf)(x) = f(x+1) - f(x)$, find $KerL$ and $ImL$
I have managed to prove that $$KerL = \{f\in P_n : f(x) = a, a \in \mathbb{R}\}.$$
Following that we have that the image of given linear transformation will be the span of
$$ \{ L(x^2), \dots, L(x^n)\},$$
that is
$$ \{ (x+1)-x, (x+1)^2 - x^2,\dots, (x+1)^n-x^n\}.$$
Obviously we have that 
$$ImL = P_{n-1},$$
but I am struggling to prove that. How would one go about it ?

Hint: It is clear that $Im L \subseteq P_{n-1}$. Use  a dimension argument to see that equality must hold. These two spaces both have dimension $n$ since the kernel of $L$ has dimension $1$. 
