egyptian fractions

egyptian fractions

can all fractions with a numerator of two be written as the sum of two different unitary fractions with mathematical proof

I assume:
\(\dfrac{2}{n}=\dfrac{1}{a}+\dfrac{1}{b} \)

if \(n\) is even then we set \(n=2k\):

\(\mathbf{\dfrac{2}{n}}=\dfrac{2}{2k}=\dfrac{1}{k}\mathbf{=\dfrac{1}{k+1} + \dfrac{1}{k(k+1)} }\)

if \(n\) is odd then we set \(n=2k-1\):

\(\mathbf{\dfrac{2}{n}}=\dfrac{2}{2k-1}\mathbf{=\dfrac{1}{k} + \dfrac{1}{k(2k-1)} }\)