Existence of 'best' and 'worst' lottery How can 'the best and worst lotteries exist when the set of outcome is finite and the rational preference relation satisfies independence axiom' be proven?
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.