There are [tex]\dbinom{|A|}k[/tex] ways of building a subset of [tex]k[/tex] elements from [tex]A[/tex], so the total number of subsets you can build is
[tex]\displaystyle\sum_{k=0}^{|A|}\binom{|A|}k[/tex]
Recalling the binomial theorem, the above sum is equal to
[tex]\displaystyle\sum_{k=0}^{|A|}\binom{|A|}k1^k1^{|A|-k}=(1+1)^{|A|}=2^{|A|}[/tex]
as required.
