Need help to understand a line of a proof of diagonalizability of real symmetric matrices

Question

I was reading a proof of diagonalizability of real symmetric matrices using the concept of generalized eigenvalues and understood all except the very starting (and fundamental) line of the proof " if a symmetric matrix is not diagonalizable then it must have generalized eigenvalues of order 2 or higher " , please explain , help me to understand , thank you .

Answer 1

A way of restating this: if a matrix $A$ is diagonalizable, then there is no vector $v$, scalar (eigenvalue) $\lambda$, and integer $n\geq2$ such that $$ (A- \lambda I)v \neq 0 \text{ but } (A-\lambda I)^nv =0 $$