Proof that first difference filter amplifies noise
Any filter whose frequency response magnitude is above unity at a given frequency $\omega$, amplifies signals at that frequency.

In discrete-time, the typical white noise is a wideband
signal whose power spectrum ranges from $-\pi$ to $\pi$.

The digital  first difference filter is an LTI system with the impulse response $h[n] = \delta[n] - \delta[n-1]$ and frequency response $H(\omega) = 1 - e^{-j\omega}$, whose magnitude is:

$$ |H(\omega)| = \sqrt{ 2 - 2 \cos(\omega) } = \sqrt{ 4 \sin^2(\omega/2) } = 2|\sin(\omega/2)|$$

as you can see as $\omega \to \pi$, then the magnitude goes to $2$ which is larger than unity and therefore amplifies any noise content found at those frequencies.

Note that, in any digital system, there will always be noise due to inherent quantization of numerical representation format, and therefore it will always be amplified using the given first difference filter.