Black-Scholes Assumptions and the American Call Delta

This is a three-part question about the Black-Scholes framework for a non-dividend-paying stock. 1. **State the key assumptions** of the Black-Scholes model. 2. **Show that an American call on a non-dividend-paying stock has the same value as the corresponding European call.** Outline a no-arbitrage argument -- why is early exercise never optimal? 3. **Derive the delta** $\Delta = \partial C / \partial S$ of a European call with strike $K$ and maturity $T$. Express your answer in terms of $N(d_1)$, and explicitly define $d_1$.