Bohr's postulate to explain stable orbits in a hydrogen atom:The stationary orbits are those orbits for which the angular momentum of the electron is an integral multiple of $\frac{\mathit{h}}{2\mathit{\pi }}$, where $\mathit{h}$ is Planck's constant.
Electron in an atom revolves because of the balance of Coulomb force of attraction between the protons and electrons and the centripetal force.
In ${\mathit{n}}^{\mathit{t}\mathit{h}}$ orbit, $\frac{\mathit{m}{\mathit{v}}_{\mathit{n}}^2}{{\mathit{r}}_{\mathit{n}}}=\frac{1}{4\mathit{\pi }{\mathit{\epsilon }}_0}\frac{{\mathit{e}}^2}{{\mathit{r}}_{\mathit{n}}^2}$
$\therefore {\mathit{v}}_{\mathit{n}}=\frac{\mathit{e}}{\sqrt{4\mathit{\pi }{\mathit{\epsilon }}_0\mathit{m}{\mathit{r}}_{\mathit{n}}}}....\left(\mathit{i}\right)$
Again, ${\mathit{r}}_{\mathit{n}}=\frac{{\mathit{\epsilon }}_0{\mathit{h}}^2{\mathit{n}}^2}{{\mathit{e}}^2\mathit{\pi }\mathit{m}}...\left(\mathit{i}\mathit{i}\right)$
Putting in equation (i)
${\mathit{v}}_{\mathit{n}}=\frac{\mathit{e}}{\sqrt{4\mathit{\pi }{\mathit{\epsilon }}_0\mathit{m}\frac{{\mathit{\epsilon }}_0{\mathit{h}}^2{\mathit{n}}^2}{{\mathit{e}}^2\mathit{\pi }\mathit{m}}}}$
$=\frac{{\mathit{e}}^2}{2{\mathit{\epsilon }}_0\mathit{h}\mathit{m}}$
$\therefore {\mathit{v}}_{\mathit{n}}\propto \frac{1}{\mathit{n}}$