Neutron decay

In case you're still a little leery about the neutrino's existence or conservation of lepton number, take a look at the decay of a neutron:
\[
n \rightarrow p + e^- + \overline{\nu}_e
\]
If lepton number conservation weren't a requirement, we could also see the reaction
\[
 n \rightarrow p + e^-
\]
These look pretty similar at first, but it turns out they're easily distinguishable. If the neutron starts at rest, then momentum conservation dictates that the momenta of the products (three in the reaction we observe, two in the second one above) sum to zero. In the second reaction, since there are only two products, they go ricocheting off in exactly opposite directions. In the first, while we are very unlikely to detect the neutrino, the electron and positron no longer have opposite momenta, which is easily detectable (assuming you start with a slow enough neutron). And it turns out (physics is, after all is said and done, an empirical science) that the first reaction occurs and the second doesn't. This demonstrates the existence of the little neutral particle, and other similar experiments allow physicists to conclude that lepton number is always conserved.