Assume the real number $a$ such that $0.5 < a < 1$ and $a$ can be expressed as an Egyptian fraction of length $l$, wich means for natural numbers $n_1$ to $n_l$ we have:
$$ a = \sum^l_1{\frac{1}{n_i}} $$ for example $a = \frac{47}{60}$ can be written as $\frac{1}{3} + \frac{1}{4} + \frac{1}{5}$ or $\frac{1}{2} + \frac{1}{4} + \frac{1}{30}$
Prove that for any $a$ in the interval $(0.5, 1)$, if an Egyptian fraction of length $l$ exists, there should also exist an Egyptian fraction of length $l$ containing $\frac{1}{2}$ in it. (Or come up with a counter-example)
P.S. My question comes from my previous question. Although I can't provide a straightforward mathematical proof, my computer evaluation (checking all possibilities) shows there is no such a number.