Return to Revisions

A result from the present site illustrates this nicely. The task is to prove the very challenging inequality $$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geqslant x^n+1$$for $x>0$, where $n$ is any natural number. The first step is to generalize it to$$\begin{equation} \left(\frac{x^a+1}{x^{a-1}+1}\right)^{a+b-1}+\left(\frac{x^b+1}{x^{b-1}+1}\right)^{a+b-1}\geqslant x^{a+b-1}+1, \end{equation}$$where $a$ and $b$ are arbitrary real numbers $\geqslant1$. Thereafter, a series of ingenious substitutions, along with yet more generalizations, enable a proof by relatively elementary mathematics.