Exercises
[0MZ] Given \(f,g\) continuous functions on \(ℝ\), we define
\[ d(f,g)=\sup _{x∈ℝ}|f(x)-g(x)|\ . \]Prove that \(d\) is a distance on \(X=C(ℝ)\), in the extended sense of the exercise [0MX].
Let \(f∼ g\iff d(f,g){\lt}∞\) as before, show that the family of equivalence classes \(\frac X∼\) has the cardinality of the continuum.
Solution 1