prove the spectral theorem for commutative operators with guidance

let $V$ be a vector space of final dimension above $F$ and $T,Scolon V to V$ diagonalizable linear operators for which holds $ST=TS$.
I need to prove $S+T$, $TS$ are also diagonalizable.

For that I have the following guidance:

I need to show that from $ST = TS$ can be infered that $T, S$ simultaneously diagonalizable, i.e., there is a base $B$ of $V$ in which both operators represented by diagonal matrices simultaneously.
That I need to show in the following way:

• firstly, I need to show that for each $lambda$ holds $S(T-lambda I)=(T-lambda I)S$

• secondly, I need to show that $W_lambda = ker(T-lambda I)$ is invariant subspace.

• finally, show by induction on the space dimension $n$ that $T, S$ simultaneously diagonalizable, i.e show that $dim(W_lambda)ledim(V)$ and to use the induction assumption to get that the reductions of $T,S$ on $W_lambda$, i.e $T_{|W_{lambda }}$, $T_{|W_{lambda }}$ simultaneously diagonalizable.

  In the induction step I need to unite the bases of the sub-spaces of $W_lambda$ which I got previously.

I easily did the first two points:

And I’ve got stuck in further attempts to continue.

Somebody helped me out with it, but I didn’t understand his solution, because he wrote statement 4, which he never mentioned before:

Can anyone help me figure it out or maybe correct the proof according to the written steps ?