The definition of convergence states that the {a_n}{an} converges if:
AA epsilon > 0 " " EE N: AA m,n>N " " |a_n-a_m| < epsilon∀ε>0 ∃N:∀m,n>N |an−am|<ε
So, given epsilon >0ε>0 take N > log_2(1/epsilon)N>log2(1ε) and m,n > Nm,n>N with m < nm<n
As m < nm<n, (2^(-m) - 2^(-n) )> 0(2−m−2−n)>0 so |2^(-m) - 2^(-n)| = 2^(-m) - 2^(-n)∣∣2−m−2−n∣∣=2−m−2−n
2^(-m) - 2^(-n) = 2^(-m)(1- 2^(m-n))2−m−2−n=2−m(1−2m−n)
Now as 2^x2x is always positive, (1- 2^(m-n) ) < 1(1−2m−n)<1, so
2^(-m) - 2^(-n) < 2^(-m) 2−m−2−n<2−m
And as 2^(-x)2−x is strictly decreasing and m > N > log_2(1/epsilon)m>N>log2(1ε)
2^(-m) - 2^(-n) < 2^(-m) < 2^(-N) < 2^(-log_2(1/epsilon)2−m−2−n<2−m<2−N<2−log2(1ε)
But:
2^(-log_2(1/epsilon) )= 2^(log_2(epsilon)) = epsilon2−log2(1ε)=2log2(ε)=ε
So:
|2^(-m) - 2^(-n)| < epsilon∣∣2−m−2−n∣∣<ε
Q.E.D.
