Of course, I have oversimplified the meaning of algebra. 🙂
Here is an example. Let me know what you think. (Reference: Algebra 3rd Edition by Serge Lang).
Let G be a commutative monoid, and be elements of G. Let be a bijection of the set of integers onto itself. Then,
Proof by mathematical induction:
PS: if one gets scared by the above notation, one can expand it and see its meaning. Try that.
It is clearly true for . We assume it for . Let k be an integer such that . Then,
Define a map of into itself by the rule:
which by induction is equal to as desired.
Some remarks: As a student, I used to think many a times that this proof is obvious. But it would be difficult to write it. I think this cute little proof is a good illustration of “how to prove obvious things in algebra.” 🙂