This was the question that was asked when it was discovered that the Rydberg formula to Hydrogen spectra coincided with the formula to solve standing wave equations. Davisson and Germer discovered that electrons could be waveforms if they were passed through a suitable difffraction grating. They used the de Broglie formula (which maps electron momentum to a wavelength in the exact same way that in classical Electrodynamics), and discovered that the atoms in a crystal such as metallic nickel formed a suitable difffraction grating. The electron gun was set up and pointed at a thin nickel film to find the predicted pattern of diffraction.
It worked. This is my favorite example of the wave nature electrons. This was the lab that I did as a student. This is the experiment I believe should be discussed first, before the confusion of the double-slit experiment.