help with proof by induction please - see below!

the power set of a finite set a, denoted p(a), is defined to be the set of all subsetsof a. for instance, if a = {1, 2}, then p(a) = {∅, {1}, {2}, {1, 2}}. note that p(a) isa set which contains sets as elements, and that the empty set, ∅, is a subset of allsets.given that a contains n elements, conjecture a formula for the number of elementsin p(a) and prove it by induction.

by trying a few small sets, we see the following:
 
|a| = 0 ---> |p(a)| = 1
 
|a| = 1 ---> |p(a)| = 2
 
|a| = 2 ---> |p(a)| = 4
 
|a| = 3 ---> |p(a)| = 8
 
this evidence suggests that maybe, for a set with n elements, the power set will have 2n elements. we prove this by induction.
 
the base case is established by the observations above. now, consider a set a with n elements, and add a single new element: call it z. let's denote a ∪ {z} as set b, which has n+1 elements. we can enumerate the subsets of b, in pairs, as follows: any subset of a is a subset of b, and any subset of a, with z added, is a different subset of b. for every subset of a, we obtain precisely two subsets of b in this way, and every subset of b is accounted for in this enumeration. thus, the number of subsets of b is precisely twice the number of subsets of a. so:
 
|p(b)| = (2)|p(a)| = (2)(2n) = 2n+1,
 
as desired.