The previous example might remind you of the racetrack principle from calculus, which says that if \(f(a) \lt g(a)\text{,}\) and \(f'(x) \lt g'(x)\) for \(x > a\text{,}\) then \(f(x) \lt g(x)\) for \(x > a\text{.}\) Same idea: the larger function is increasing at a faster rate than the smaller function, so the larger function will stay larger. In discrete math, we don't have derivatives, so we look at differences. Thus induction is the way to go.
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