Well, the point is that although the method of mathematical induction can be useful and is useful for proving certain statements, I somehow always like things to be proved in some other way than with the mathematical induction, because, to me, the method of mathematical induction is not too much reliable method.
But let us skip now my feelings about mathematical induction and let us get to what I would like to know at the moment.
My question is:
Does there exist at least one example of the mathematical fact (or truth) that can be proven with mathematical induction and for which it is not known does there exists some other method with which we could prove it?
This question is motivated by the fact that whatever mathematical fact I have seen to be proved with mathematical induction I have also seen it been proved in some other way, so I would like to know if someone knows about some mathematical fact that can be proven with mathematical induction but resists attempts to be proven with some other method that does not use mathematical induction in any way?