A charge +q is placed at $(a,0,0)$ and another $+q$ charge is placed at $(-a,0,0)$. A charge $-q$ ${}_1$ is placed at the origin. If it is slightly displaced along y axis. Then :

Let the charge be displaced to $(0,y)$.

The electric force due to a point charge is given by $F=\frac{qQ}{4\pi {ϵ}_0{r}^3}\vec{r}$

The electric force due to the charge at $(a,0,0)$ is $F_1=\frac{1}{4\pi {ϵ}_0}\frac{-q{q}_1}{(\sqrt{a^2+y^2}{)}^3}(-a\hat{i}+y\hat{j})$

The electric force due to the charge at $(-a,0,0)$ is $F_2=\frac{1}{4\pi {ϵ}_0}\frac{-q{q}_1}{(\sqrt{a^2+y^2}{)}^3}(a\hat{i}+y\hat{j})$

Thus, the total force experienced by $q_1$ due to the system of charges is

$F=F_1+F_2=\frac{1}{4\pi {ϵ}_0}\frac{-q{q}_1}{(\sqrt{a^2+y^2}{)}^3}(2y\hat{j})$

As the displacement is small, i.e., $y<<a$, $F\approx \frac{1}{4\pi {ϵ}_0}\frac{-q{q}_1}{a^3}(2y)\hat{j}=-\frac{q{q}_1}{2\pi {ϵ}_0{a}^3}y\hat{j}$

We get  $F\propto -y$

The force is proportional to the displacement and is in the opposite direction. Thus, it executes a SHM.