In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$. By using spectral resolutions of $A$, $B$ and $C$, show that if the strictly positive integers $j$ and $k$ satisfy $j + k \leqslant n + 1$, we have $$c_{j+k-1} \leqslant a_{j} + b_{k}.$$ Deduce that for every integer $j$, $1 \leqslant j \leqslant n$, $$a_{j} + b_{n} \leqslant c_{j}.$$
In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
By using spectral resolutions of $A$, $B$ and $C$, show that if the strictly positive integers $j$ and $k$ satisfy $j + k \leqslant n + 1$, we have
$$c_{j+k-1} \leqslant a_{j} + b_{k}.$$
Deduce that for every integer $j$, $1 \leqslant j \leqslant n$,
$$a_{j} + b_{n} \leqslant c_{j}.$$