Specific case of Jordan normal form

I'm having trouble understanding Jordan normal form for matrices. I can prove it works for 3x3, but the general proof is a bit over my head, so I figured I'd try the 4x4 case. But even the really easy case, where matrix A has a single eigenvalue x and eigenspace of dimension 1, I can't prove it.

Obviously the Jordan normal form should look like:

x 1 0 0

0 x 1 0

0 0 x 1

0 0 0 x

but I'm not sure how to prove an arbitrary 4x4 matrix meeting those criteria can be put in this form.