6. Prove that each of the following functions f has properties (1), (2), and (3) in Proposition 2.4.3: (a) f (x ) := x/(1 + x ); (b) f (x ) := tan-1 x ; (c) f (x ) := min(x, 1), 0 ≤ x ∞.

7. Show that the functions x i→ sin(nx ) on [0, 1] for n = 1, 2,... , are not equicontinuous at 0.

8. A function f from a topological space (S, T ) into a metric space (Y, d) is called bounded iff its range is bounded. Let Cb (S, Y, d) be the set of all bounded, continuous functions from S into Y . For f and g in Cb (S, Y, d) let dsup( f, g) := sup{d( f (x ), g(x )): x ∈ S}. If (Y, d) is complete, show that Cb (S, Y, d) is complete for dsup.