I'm reading a paper that made a certain assumption as being trivial (I believe), but which I set out to prove. And now I'm kind of stuck.
Let $I=[a,b]$ and $\bf{x}$ $= \{ x_1, x_2, \ldots, x_N \} \in I$. In other words, $\bf{x}$ is a finite collection of points on interval $I$. Now let $L=[\text{inf}(\bf{x}$) $,\text{sup}(\bf{x}$ $)]$. Now let $\bf{\lambda } = \{\lambda _1, \lambda _2, \ldots, \lambda _N\}$; $\lambda _i \geq 0$, s.t. $\sum_{i=1}^{N} \lambda_i = 1$.
Is there a way to show that $\sum_{i=1}^{N} \lambda_i x_i \in L$, i.e. $\text{inf}(\bf{x}$) \leq \sum_{i=1}^{N} \lambda_i x_i \leq \text{sup}(\bf{x}$)$?
Is it even true?