In mathematics, the inequality
is not hard to prove with quadratic root discriminant.
But the problem is that my computer science roommate want me to give an induction method. So I tried a new method to prove it, in which I used a small trick deserving taking a note.
Let me firstly restate the problem here: it’s known that
, prove that
The equality holds if and only if
the inequality obviously holds true.
Assume that when
the inequality holds true.
Therein lies the problem, we cannot break the formula into two parts equally if induction is used.
Instead of proving this inequality directly, we consider another problem which has a subtle link with the issue we are dealing with. Say, when
Actually, by induction method, because
this statement holds true definitely. Further more, it gives
Otherwise, the new problem value would be even smaller.
The inspring point for this method is to add a free variable to cater the old problem, though technical when applying it. And I hope this trick will bring me more stuff.