In the book, Linear Algebra Done Wrong , Sergey Treil says that one can notice that the algebraic multiplicity of an eigenvalue $\lambda$ is greater than its depth (minimum $k$ such that $(A-\lambda I)^k|_{E_\lambda} = 0$, where $E_\lambda$ is the generalized eigenspace of $\lambda$). How exactly does one conclude this?
Note that we are building towards Jordan canonical form, so we can't use that here.
Here is a screenshot:
You have to convince yourself that the depth is the multiplicity of $\lambda$ in the minimal polynomial. Once you have that, you can conclude by saying that the minimal polynomial divides the characteristic polynomial.
To prove the first statement, if $d_\lambda$ is the depth of $\lambda$, then $\prod_\lambda (X-\lambda)^{d_\lambda}$ annihilates $A$, so it is divisible by the minimal polynomial. If $m_\lambda$ is the multiplicity of $\lambda$ in the minimal polynomial, we get that $d_\lambda\geq m_\lambda$ for all $\lambda$.
To prove the other inequality, you just have to prove that $(A-\lambda I)^{m_\lambda}_{E_\lambda}=0$.
Write $\mu_A=(X-\lambda)^{m_\lambda} P,$ where $P(\lambda)\neq 0$. Then $P(A)_{E_\lambda}$ is invertible ($\lambda$ is not a root of $P$) and since $\mu_A(0)=0$, it i s also $0$ on $E_\lambda$ and we get what we want.