How do I prove this equation for any function $f$ (can this even be proven for an arbitrary function?), where $a$ and $a^{\dagger}$ are the creation and annihilation operators for the 1D harmonic oscillator?

$$af(a^{\dagger}a) = f(a^{\dagger}a + 1)a$$

I have tried expanding $f$ as a power series and using the commutation relation of $a$ and $a^{\dagger}$.

How do I prove this equation for any function $f$ (can this even be proven for an arbitrary function?), where $\hat a$ and $\hat a^{\dagger}$ are the creation and annihilation operators for the 1D harmonic oscillator?

$$\hat af(\hat a^{\dagger} \hat a) = f(\hat a^{\dagger} \hat a + 1) \hat a$$

I have tried expanding $f$ as a power series and using the commutation relation of $\hat a$ and $\hat a^{\dagger}$.

How do I prove this equation for any function $f$ (can this even be proven for an arbitrary function?), where $a$ and $a^{\dagger}$ are the creation and annihilation operators for the 1D harmonic oscillator?

$af(a^{\dagger}a) = f(a^{\dagger}a + 1)a$

I have tried expanding $f$ as a power series and using the commutation relation of $a$ and $a^{\dagger}$.

How do I prove this equation for any function $f$ (can this even be proven for an arbitrary function?), where $a$ and $a^{\dagger}$ are the creation and annihilation operators for the 1D harmonic oscillator?

$$af(a^{\dagger}a) = f(a^{\dagger}a + 1)a$$

I have tried expanding $f$ as a power series and using the commutation relation of $a$ and $a^{\dagger}$.