Let $x_1, \dots, x_N \in \mathbb{R}$ each be a
D993: Real number.
Let $\lambda_1, \dots, \lambda_N \in [0, \infty)$ each be an
D4767: Unsigned real number such that
| (i) |
\begin{equation}
\sum_{n = 1}^N \lambda_n = 1
\end{equation}
|
Denote $M : = \max(x_1, \ldots, x_N)$ and $m : = \min(x_1, \ldots, x_N)$. We have
\begin{equation}
\lambda_1 x_1 \leq \lambda_1 M, \quad
\ldots, \quad
\lambda_N x_N \leq \lambda_N M
\end{equation}
Applying
R1904: Isotonicity of finite real summation, we therefore get
\begin{equation}
\sum_{n = 1}^N \lambda_n x_n
\leq \sum_{n = 1}^N \lambda_n M
= M \sum_{n = 1}^N \lambda_n
= M
\end{equation}
In the other direction, we have
\begin{equation}
\lambda_1 x_1 \geq \lambda_1 m, \quad
\ldots, \quad
\lambda_N x_N \geq \lambda_N m
\end{equation}
whence we obtain
\begin{equation}
\sum_{n = 1}^N \lambda_n x_n
\geq \sum_{n = 1}^N \lambda_n m
= m \sum_{n = 1}^N \lambda_n
= m
\end{equation}
$\square$
