<s>[INST] Prove something similar to a variant of Cauchy-Shwarz inequality Cauchy-Shwarz Inequality is:
$$(a_1b_1 + a_2b_2 + \cdots + a_nb_n)^2 \leq (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + \cdots + b_n^2)$$
However, it can be manipulated as:
$$\sqrt{(a_1-b_1)^2 + (a_2-b_2)^2 + \cdots + (a_n-b_n)^2} \leq \sqrt{a_1^2 + a_2^2 + \cdots + a_n^2} + \sqrt{b_1^2 + b_2^2 + \cdots + b_n^2}$$
I'm tasked with proving the following inequality:
$$\sqrt{(a_1 + b_1 + \cdots + z_1)^2 + (a_2 + b_2 + \cdots + z_2)^2 + \cdots + (a_n + b_n \cdots + z_n)^2} \leq \sqrt{a_1^2 + a_2^2 + \cdots + a_n^2} + \sqrt{b_1^2 + b_2^2 + \cdots + b_n^2} + \cdots + \sqrt{z_1^2 + z_2^2 + \cdots + z_n^2}$$
I've proved both Cauchy-Shwarz and its manipulation, but am lost when it come to the inequality right above. Hints and/or solutions are welcome.
 [/INST] A geometrical interpretation would be the following:
Consider an $m$-dimensional box and subdivide it in $n\times n\times\dots\times n$ grid. Then its diagonal length is at most the sum of the diagonals in a "diagonal chain" of boxes of the grid.
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