Problem 3.a: Give an example of a continuous function f: [0,1] -> R and a Cauchy sequence {n} in [0,1] such that (f(n)) is not a Cauchy sequence.
b: Show that if f: A -> R is uniformly continuous and {n} ⊆A is a Cauchy sequence, then (f(x_n)) is also a Cauchy sequence.
Extra notes: (b) can be used to show that f is uniformly continuous on [a,b] if and only if f can be extended to a continuous function on [a,b]. Use this to easily show that x*sin(1/x) is uniformly continuous on (0,1)!
