What is the significance of dividing the euclidean metric by $\sqrt{n}$?

Question

From Munkres Topology:

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In the above proof, I was trying to figure out where Munkres obtained the $\sqrt{n}$ and I think it was by dividing the sum of the squares by $n$ in the euclidean metric. Is this correct?

However, now I have the following concern:

What is the significance of the root mean square of the euclidean metric?

$$\left(\frac{\sum_{i=1}^n (x_i-y_i)^2}{n}\right)^{1/2}$$

At first I thought it was probably the average of the distances between the coordinates but when I calculated an example in two dimensions, I saw this was not the case. For example, consider the points $(1,2)$ and $(7,11)$. We see $\sqrt{6^2+9^2}/2=\sqrt{117}/2 = 7.6485$. But the average of $6$ and $9$ is $7.5$.

So, what is the significance of dividing the euclidean metric by $\sqrt{n}$? What does this value represent?

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I understand the heart of the proof, that is, why the topologies are the same. What I am asking is more of an intuitive question. What the calculation behind the proof means.

Answer 1

$\sqrt n$ is just the length of the diagonal of the $n$-cube: the above inequality just says that $$ \max_i \{ |y_i - x_i| \} \leq \sqrt {\sum_i (y_i - x_i)^2} \leq \sqrt {\sum_i (\max_i \{ |y_i - x_i |\})^2} = \sqrt {n (\max_i \{ |y_i - x_i |\})^2} = $$ $$ = \sqrt n \max_i \{ |y_i - x_i| \} $$