## The suprema metric and unit distances

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 $$[0, 1] \to \mathbb R$$.
Show that $$(X, \rho_s)$$ is a metric space.
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 $$i \neq j$$.