<s>[INST] 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.
 [/INST] Hint $\ $ For $\rm\ A = 1/a,\ B = 1/b,\ C = 1/c\ $ it becomes obvious, viz.
$$\rm C = \frac{A+B}2\ \Rightarrow\ A \ge C \ge B\ \ or\ \ B \ge C \ge A$$ 
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