Proof Step 1 - First Activity

Let $g_1$ be greedy's first activity (ends earliest) and $o_1$ be OPT's first activity.

Case $1$: $o_1 = g_1$. No change needed. Both agree on the first activity.

Case $2$: $o_1 \neq g_1$. Since $g_1$ ends earliest, $end(g_1) \leq end(o_1)$. Replace $o_1$ with $g_1$ in OPT to create OPT'. Is OPT' valid? Yes! Since $g_1$ ends no later than $o_1$, it cannot conflict with OPT's second activity (which starts after P10 ends). OPT' has the same count as OPT.