Hi, I have four questions questions (proof by induction is prefered,
but other methods will do):

1. Show that {0, 1}* is computably enumerable.

2. Prove that an infinite language is computably enumerable if and
only if there is a computable function that enumerates it without
repetition.

3. Prove that a set is decidable if and only if there is a one-to-one
computable function which enumerates its elements in increasing order.

4. Prove that a function f is computable if and only if it's graph
{(x, f(x))} is computably enumerable.

Left as an exercise for the student... :-)