Proof for composed signatures
Sketch.

Suppose you have an adversary against the composed scheme with advantage $\epsilon_3$. Observe that whenever you have a signature for the composed system, you have a signature for each of the three component signature schemes.

It should then follow that a forger for the composed signature scheme can be turned into a forger for each of the three signature schemes, and it will have advantage $\epsilon_3$. (This involves a few easy technicalities.)

With that proved, it follows that we can bound the adversary's advantage $\epsilon_3$ by the advantages $\epsilon$, $\epsilon'$ and $\epsilon''$.