The suprema metric and unit distances
Task number: 3168
Let \(X\) be the set of all bounded real functions on \([0, 1]\). The suprema metric \(X\) is defined by \[ \rho_{s}(f, g) = \sup \{|f(x) - g(x)|\colon x \in [0, 1]\}, \] where \(f\) and \(g\) are bounded functions on \([0, 1] \to \mathbb R\).
Variant 1
Show that \((X, \rho_s)\) is a metric space.
Variant 2
Find an infinite number of functions \(f_n\colon [0, 1] \to \mathbb R\) for \(n \in \mathbb N\) such that \(\rho_s(f_i, f_j) = 1\), when \(i \neq j\).
