The three fractions sum to a positive value (they only need sum to a non-negative value for the inductive proof).
Easiest is..
and as the fraction on the right has a bigger denominator, then it is less than the fraction on the left,
hence the subtraction yields a positive answer.
What this then means is....
if P(k) really is valid, P(k+1) will also be valid.
This is because if the sum of any number of k terms is >13/24,
then the sum of k+1 terms is greater again.
Hence, since the sum of 2 terms is >13/24, the sum of any (>2) number of terms is >13/24.