To illustrate this point, let me here state an interesting result, also called Bertrand' s Postulate and discovered by Joseph Bertrand in 1845. However, verification, no matter for how many cases, does not constitute proof. In fact Bertrand failed to come up with a proof to this apparently true statement. The first proof was due to the Russian mathematician Pafnuty Chebyshev in about 1850. His proof was rather difficult, but a much simpler proof was given by the Indian mathematical genius Ramanujan.
Several observations derive from the above.
One is that mathematics is a human activity. A second is that much of it is done through trial and error, that is, experimentation. The historical aspect of the subject is also revealed here, and so on. The problem with some of us teachers is that, more often than not, we tend to present mathematics to students in its final form. We hide the hard work and fun that goes into coming up with the final argument.