The topologies on $\displaystyle \mathbb{R}^n$ induced by the euclidean metric d and the square metric rho are the same as the product topology on $\displaystyle \mathbb{R}^n$.

Euclidean metric

$\displaystyle d(x,y)=\sqrt{(x_1-y_1)^2+\cdots +(x_n-y_n)^2}$

Square metric

$\displaystyle \rho(x,y)=\max\{|x_1-y_1|,\cdots ,|x_n-y_n|\}$

So proof says:

Let $\displaystyle x=(x_1,\cdots ,x_n)$ and $\displaystyle y=(y_1,\cdots ,y_n)$ be two points in $\displaystyle \mathbb{R}^n$. It is simple algebra to check that

$\displaystyle \rho(x,y)\leq d(x,y)\leq\sqrt{n}\rho(x,y)$

(1) Where and why did they have this inequality?

(2) I know it is stated that it is simple algebra to check it, but I don't see it. How is checked?