Problem 1
Given a metric space \((X, p)\), prove that a) \(|\rho(x, z)-\rho(y, u)| \leqslant \rho(x, y)+\rho(z, u) \quad(x, y, z, u \in X)\); b) \(|\rho(x, z)-\rho(y, z)| \leqslant \rho(x, y) \quad(x, y, z \in X)\).
Problem 2
Verify that $$ \left(\sum_{k=1}^{n} a_{k} b_{k}\right)^{2}=\sum_{k=1}^{n} a_{k}^{2} \sum_{k=1}^{n} b_{k}^{2}-\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n}\left(a_{i} b_{j}-b_{i} a_{j}\right)^{2} $$ Deduce the Cauchy-Schwarz inequality (3) from this identity.
