I forgot to answer the rest of this....
it isn't a matter of certain values of "m" making the inequality true.
You simply sum all the fractions (increasing m by 1 in each fraction) until m reaches n (not necessary when proving by induction).
For all n greater than or equal to 2, your sum ought to be greater than 13/24.
The general idea of induction is as follows: (1) I show that something is true for a specific value of n, e.g. n=2. Then (2) I assume that this same statement is true for any n and (3) I show that it is then true for n+1. It follows that it must be true for any value since I can start by e.g. n=2 => true for n+1 = 3 => true for n+2 = 4 => ...
So we showed that it is true for n=2 already. Now we assume that it is true for
Thus we can solve
Thus:
and we completed our proof by induction since we can now construct any n as shown above.
This last step is true if and only if This is all that is left to show to finish the proof. To do so, simply get a common denominator. If both numerator and denominator of the new, "merged" term are positive, you are done.