How did Hertz prove existence of radio waves?
How did Prof. Hertz know that the sparks appearing in his remote receiver were not caused (1) by induction (as described by Faraday) or (2) by what today is called "near-field radiation?"


The spark at the receiver in response to a transmission from a spark gap transmitter was only the initial observation that inspired Hertz to do more conclusive experiments.

Hertz's experiments in 1886 to 1889 were explicitly an attempt to verify Maxwell's theories, introduced a decade earlier. Therefore he had an idea of the phenomena he was attempting to observe, and he knew the likely velocity of the waves, measured by Foucault in 1862.

Per Wikipedia


  In the apparatus Hertz used, the electric and magnetic fields would radiate away from the wires as transverse waves. Hertz had positioned the oscillator about 12 meters from a zinc reflecting plate to produce standing waves. Each wave was about 4 meters long. Using the ring detector, he recorded how the wave's magnitude and component direction varied. Hertz measured Maxwell's waves and demonstrated that the velocity of these waves was equal to the velocity of light. The electric field intensity, polarity and reflection of the waves were also measured by Hertz. These experiments established that light and these waves were both a form of electromagnetic radiation obeying the Maxwell equations. 


The main feature that distinguishes these waves from near-field or purely inductive effects, is the observation of standing waves from the transmitter placed in front of a reflector.