Evidence that proves that electrons move around in definite pathways around nucleus is the emission spectra of the elements.

When the emission spectra of the elements was observed, there were only a certain frequencies and not a full color spectrum. This proves that electrons can only emit some specific energy frequencies only.

As electrons emit these frequencies when they move from a higher energy level to a lower energy level, it is evident that electrons jump between specific energy levels or pathways only and can’t exist anywhere in between.