''Motion of a simple pendulum from the mean position for small displacement is a simple harmonic motion'' - Explain this statement.

(N/A) simple pendulum consists of a bob of mass $m$ suspended by a light string of length $l$. When the pendulum is displaced by a small angle $\theta$ from the mean position,the restoring force is provided by the tangential component of gravity.
$1$. The restoring force is given by $F = -mg \sin \theta$.
$2$. For small angles,$\sin \theta \approx \theta$ (in radians),where $\theta = \frac{x}{l}$ ($x$ is the linear displacement).
$3$. Substituting this,we get $F = -mg \left( \frac{x}{l} \right) = -\left( \frac{mg}{l} \right) x$.
$4$. Since $m, g,$ and $l$ are constants,the restoring force $F$ is directly proportional to the negative displacement $x$,i.e.,$F \propto -x$.
$5$. This is the defining condition for Simple Harmonic Motion $(SHM)$. Thus,the motion is simple harmonic with an angular frequency $\omega = \sqrt{\frac{g}{l}}$.