Step in proof of Cayley-Hamilton theorem in Steinberg's book

Question

I am reading "Representation Theory of Finite Groups - An Introductory Approach" by Benjamin Steinberg, and making exercise 2.9. I can unfortunately not find a solution anywhere. Most of the steps I have figured out, except step 4. The exercise is the following:

Exercise 2.9. This exercise sketches a proof of the Cayley–Hamilton Theorem using a little bit of analysis.

1. Use Exercise 2.8 (prove that for every complex $n\times n $ matrix $A$ there exists an invertible matrix $P$ such that $P^{-1}AP$ is upper triangular) to reduce to the case when A is an upper triangular matrix.
2. Prove the Cayley–Hamilton theorem for diagonalizable operators.
3. Identifying $M_{n}(\mathbb{C})$ with $\mathbb{C}^{n^{2}}$ , show that the mapping $Mn(\mathbb{C}) \rightarrow M_{n}(\mathbb{C})$ given by $A \mapsto p_{A}(A)$ is continuous. (Hint: the coefficients of $p_{A}(x)$ are polynomials in the entries of $A$.)
4. Prove that every upper triangular matrix is a limit of matrices with distinct eigenvalues (and hence diagonalizable).
5. Deduce the Cayley–Hamilton theorem.

If anyone could help with this or give some kind of hint it would be greatly appreciated. I just don't understand how I'm supposed to come up with such a sequence. I was considering maybe it's possible to use proof by contradiction and generalized eigenvectors, is this of any use?