Unipotent matrix similar to an upper-triangular matrix

"Any unipotent matrix is similar to an upper-triangular matrix with 1's on the diagonal"...This is usually alleged, but I have no idea how to demonstrate that, starting with the definition : $A$ is unipotent if and only if there is $k\in \mathbb{N}$ so that $(A-I_n)^k=0$.And I browsed Internet for hints but found nothing useful. I am not looking here for a ready-made solution, but I would like to understand what is the procedure, what are the steps one has to make, in order to proceed from definition to the result I stated above.Thanks in advan...Read more