I'm reading Hoffman and Kunze linear algebra book and in the end of the page 204, they said:
The theorem 5 is the following:
I didn't understand why we can use the theorem 5 to prove the Cayley-Hamilton theorem, i.e., if $p_c$ is the characteristic polynomial, then $p_c(T)=0$.
We can easily see that
Where $p_m$ is the minimal polynomial.
However I'm having troubles to understand why $l_i\le d_i$.