0! = 1 can be proved using many methods. While some use patterns or logic, giving reasons for mathematical consistency & this & that, I shall use calculus here.
We have the gamma function denoted by , which is defined by
This function arose while solving an interpolation problem. The problem was to find a monotonic function defined over which took the value at . It can be solved by evaluating the above improper integral which converges for .
The gamma function is defined for the whole of real line provided we take for .
We are interested in the case when that is, a positive integer.
Integrating by parts, we have
Notice that the integral is nothing but . Thus we have, .
Repeating the above process,
This must imply that and we have already proved that . Thus,