Given $a_1 \geq a_2 \geq \dots \geq a_{k+l} \geq 0,$ prove the shifting inequality $$\sqrt{a_{l+1}^2 + \dots + a_{l+k}^2} \leq c_{k,l}(a_1 + \dots + a_k),~~~~~\textrm{where}~~c_{k,l}:=\max\left( \frac{1}{\sqrt{k}} , \frac{1}{\sqrt{4l}} \right).$$
This is from Foucart and Rauhut's book, page 173. Any help in solving this is much appreciated.
First if $k\le l$, then $c_{k,l}=\frac{1}{\sqrt{k}}$ and $c_{k,l}(a_1+\cdots +a_k)\ge \frac{1}{\sqrt{k}}ka_l\ge \sqrt{a_{l+1}^2+\cdots a_{l+k}^2}$.
Now if $k\ge l+1$, then the inequality is equivalent to $c_{k,l}^2(a_1+\cdots +a_k)^2\ge a_{l+1}^2+\cdots a_{l+k}^2$, and it remains to prove that $$c_{k,l}^2((l+1)a_{l+1}+a_{l+2}+\cdots +a_k)^2\ge a_{l+1}^2+\cdots +a_{k-1}^2+(l+1)a_k^2$$ Since LHS-RHS is a concave function with respect to $a_{l+2},\cdots a_{k-1}$, we only need to prove $$c_{k,l}^2((l+1)a_{l+1}+(k-l-1)a_k)^2\ge (a_{l+1}^2+ka_k^2)$$ and $$c_{k,l}^2((k-1)a_{l+1}+a_k)^2\ge ka_{l+1}^2+a_k^2$$ Can you get it from this?