Discovery of elements
The notion of chemical element in modern sense arised at the end of 19 century.

Before this, the term "element" was obviously too fuzzy to discuss here, as well as methods and proofs involved.

After this, the optical spectroscopy was method of choice to find new elements and prove their existence, 
because of distinct and reproducible spectra.
See the history of helium, for example.

Of course, besides of spectroscopy, chemists proved the finding of new elements by separation of individual substances.
For example, metals are expected to form oxides and salts with (at least roughly) expected properties and mutual transformations.
In particular, salts, even in microamounts, may be recrystallized to form visually distinct crystals of individual substance.
If such a substance possessed properties not seen before, this was indication/proof of the presence of new element.