Bound the smallest variable, then list cases

A bigger bottom number cuts the whole into more, smaller pieces, so $x\ge y$ makes $\frac{1}{x}$ no bigger than $\frac{1}{y}$, and the sum is at most $\frac{1}{y}+\frac{1}{y}=\frac{2}{y}$.

$\frac{1}{3}$ is less than the sum and the sum is at most $\frac{2}{y}$, so $\frac{1}{3}<\frac{2}{y}$.

$\frac{1}{3}<\frac{2}{y}$ means two pieces of size $\frac{1}{y}$ beat two sixths, which happens only when $y<6$.