I have a set of scalars $\{\lambda_i\}_{i=1}^{n}$ and a set of nonnegative scalars $\{\alpha_i\}_{i=1}^{n}$ such that $\sum_{i=1}^n \alpha_i = 1$. Let $\lambda_{\min}$ and $\lambda_{\max}$ denote the min and max of $\{\lambda_i\}_{i=1}^{n}$, respectively. Is it then true that $$ \sum_{i=1}^n \alpha_i \lambda_i \in [\lambda_{\min}, \lambda_{\max}]? $$
If so, how does one show this?
Since we have that $\sum_{i = 1}^{n}\alpha_i = 1$, we find: \begin{equation} \lambda_{\text{min}} = \sum_{i = 1}^{n}\alpha_i \lambda_{\text{min}} \end{equation} If we now use that $\lambda_{\text{min}} \leq \lambda_i$ for all $i \in \{1, \ldots, n\}$, we find \begin{equation} \sum_{i = 1}^{n}\alpha_i \lambda_{\text{min}} \leq \sum_{i = 1}^{n}\alpha_i \lambda_i \end{equation} and after combining we find that $$ \lambda_{\text{min}} \leq \sum_{i = 1}^{n}\alpha_i \lambda_i.$$
The same can be done for $\lambda_{\text{max}}$.