How do I prove that any unit fraction can be represented as the sum of two other distinct unit fractions? A number of the form $\frac{1}{n}$, where $n$ is an integer greater than $1$, is called a unit fraction.
Noting that
$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$
and
$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$,
find a general result of the form
$\frac{1}{n} = \frac{1}{a} + \frac{1}{b}$
and hence prove that any unit fraction can be expressed as the sum of two other distinct unit
fractions.

$$\frac{1}{N+1}+\frac{1}{N(N+1)}=\frac{N}{N(N+1)}+\frac{1}{N(N+1)}=\frac{N+1}{N(N+1)}=\frac{1}{N}.$$

Here's a question for further investigation: is the above decomposition of a unit fraction into a pair of distinct unit fractions unique? After all, there is more than one way to split a unit fraction into a triplet of unit fractions.
$$\frac12=\frac14+\frac16+\frac{1}{12},$$
but also
$$\frac12=\frac13+\frac18+\frac{1}{24}.$$
