How can I prove that for real numbers $x_k$ and $y_k$,
$x_1^k+x_2^k+ \dots + x_n^k=y_1^k+y_2^k+ \dots + y_n^k$ $(k=1,2,\dots,n)$,
where $x_1\le x_2 \le \dots \le x_n$ and $ y_1\le y_2 \le \dots \le y_n$
implies $x_k=y_k (k=1,2,\dots,n)$?
I considered the polynomial with roots $x_k$, $(t-x_1)(t-x_2)\dots(t-x_n) $and its coefficients, but could not proceed.
It follows from Newton's identities and from your hypothesis that, if $\sigma_1,\ldots,\sigma_n$ are the elementary symmetric polynomials, then$$(\forall k\in\{1,\ldots,n\}):\sigma_k(y_1,\ldots,x_n)=\sigma_k(y_1,\ldots,y_n).$$Therefore$$(X-x_1)\ldots(X-x_n)=(X-y_1)\ldots(X-y_n),$$and this means that $\{x_1,\ldots,x_n\}=\{y_1,\ldots,y_n\}$.