Prove inequality - Convex problem

Question

Let $\{q_i\}_{i=1}^n$ be a sequence such that $0\le q_1\le q_2\le \cdots\le q_n$. How can I prove that $$\sum_{i=1}^n q_is_i\le q_n,$$ where $\sum\limits_{i=1}^n s_i = 1$ for all $s_i\ge 0$?

Answer 1

The sequence satisfies $q_i\leq q_n\;\forall\, i\in\{1,\,\ldots,\,n\}$.

Now consider these manipulations: $$\sum_{i=1}^n q_i s_i\leq\sum_{i=1}^n q_n s_i=q_n\sum_{i=1}^n s_i=q_n.$$