A particle is monitored from two reference frames, S_1 and S_2 ....

Let the acceleration of frame $S_1$ with respect to an inertial frame be $\vec{A}_1$.

Given that the acceleration of frame $S_2$ with respect to $S_1$ is $\vec{a}$, the acceleration of frame $S_2$ with respect to the inertial frame is $\vec{A}_2 = \vec{A}_1 + \vec{a}$.

The pseudo force on a particle of mass $m$ in frame $S_1$ is $\vec{F}_1 = -m\vec{A}_1$.

The pseudo force on the particle in frame $S_2$ is $\vec{F}_2 = -m\vec{A}_2 = -m(\vec{A}_1 + \vec{a})$.

Subtracting the two equations yields $\vec{F}_2 - \vec{F}_1 = -m\vec{a}$.

Since frame $S_2$ is accelerating relative to $S_1$, $\vec{a} \neq 0$.

This implies $\vec{F}_2 - \vec{F}_1 \neq 0$, meaning $\vec{F}_1$ and $\vec{F}_2$ can never be equal.

Therefore, it is impossible for both $F_1$ and $F_2$ to be simultaneously zero.