Kelly versus expected wealth

(a) Prove Proposition 3.3(b): for any fixed $f \ne f^*$ in $[0, 1)$, $W_n(f^*)/W_n(f) \to \infty$ almost surely. (b) For $p = 0.55$, $b = 1$, $n = 10$: compute $E[W_{10}]/W_0$ and $P(W_{10} > 0)$ for $f = 1$, and $E[W_{10}]/W_0$ and the median of $W_{10}/W_0$ for $f = f^*$. (c) Explain in two sentences why maximising expected wealth is the wrong objective for a trader who bets repeatedly.