Robustness and its limits

(a) Prove Proposition 9.6. (b) Show that the result fails for a short bull call spread (short call $K_1$, long call $K_2>K_1$) hedged at $\sigma_h$ even when $\sigma_t<\sigma_h$ always. Identify the region of spot where the hedged short spread loses money. (c) A trader sells weekly Nifty strangles at 13% implied and argues: "realised has been below 11% for six months, so by El Karoui–Jeanblanc–Shreve I cannot lose." Find two separate flaws.