Metric spaces

Prove that in the cartesian metric space $(E, \rho)$ of the metric spaces $(E_i, \rho_i), \; \; i=1, \dots, n$ hold the relations:

i.   $\rho_i(x_i, y_i) \leq \rho(x, y)$
ii.  $\displaystyle \rho(x, y) \leq \sum_{i=1}^{n} \rho_i (x_i, y_i) $

Solution:



i. For all $i \in \{ 1, \dots, n \}$ we have that:

$$\rho_i(x_i, y_i) = \sqrt{\rho_i^2 (x_i, y_i)} \leq \sqrt{ \sum_{i=1}^{n} \rho_i^2 (x_i, y_i)  } = \rho(x, y)$$

ii. This relation follows immediately from the known inequality:

$$\sum_{i=1}^{n} a_i^2 \leq \left( \sum_{i=1}^{n} |a_i|^2 \right)^2$$

if someone subs $a_i = \rho_i (x_i, y_i ) $.