Existence of 'best' and 'worst' lottery
Hint
Let $\{1,2,\dots,N\}$ be the set of outcomes. Without loss of generality, let $1\succsim 2\succsim\cdots\succsim N$. Let $\mathbf e_i$ denote the degenerate lottery that assigns probability $1$ on outcome $i$ and $0$ on the rest. Note that any lottery $L=(p_1,\dots,p_N)$ can be written as a compound lottery $p_1\mathbf e_1+\cdots+p_N\mathbf e_N$.
Since lottery $\mathbf e_i$ amounts to getting outcome $i$ for sure, we must have
\begin{equation}
\mathbf e_1\succsim \mathbf e_2\succsim \cdots \succsim \mathbf e_N.\tag{1}
\end{equation}
Successively applying the independence axiom to $(1)$, you'll obtain $\mathbf e_1$ as (one of) the best lottery and $\mathbf e_N$ as (one of) the worst.