Before I do proofs via the 3-step mathematical induction, the classic Charlie Chaplin movie clip often comes to my mind. It nudges me to seek better ways. For example, to prove
Instead of applying the 3-step mathematical induction, let us look at Fig. 1.
If A denotes the total area of the blue rectangles, Fig.1 shows
It also shows that
A < the area under the curve .
By the fundamental theorem of calculus,
This proof suggests another inequality: