If $ \frac{1}{c} = \frac{1}{2a} + \frac{1}{2b} $, then either $ a \leqslant c \leqslant b$ or $ b \leqslant c \leqslant a $ For $a, b = 1, 2, 3, \cdots$, let $ \frac{1}{c} = \frac{1}{2a} + \frac{1}{2b} $. Then prove that either $ a \leqslant c \leqslant b$ or $ b \leqslant c \leqslant a $ holds.

$c\geq a\Rightarrow 1/2a+1/2b\leq 1/a\Rightarrow1/2b\leq1/2a\Rightarrow 1/2a+1/2b\geq1/b\Rightarrow c\leq b$. Similarly do when $c\leq a$
