Proofs are a logical sequence of conclusions. Unless you are writing a book, there is no need to concern yourself with much formality.
Here's a silly example of sort-of induction and sort-of contradiction.
My intent is to prove that all Whole Numbers are interesting.
0 - is an interesting number, since it has unusual additive and multiplicative properties
1 - is an interesting number, at least since it is odd but is not prime.
2 - is an interesting number at least because it is even and it is prime.
3 - is an interesting number at least because it is the lowest odd prime.
This demonstrates that there are interesting Whole Numbers and that there are no non-interesting Whole Numbers less than these.
If we assume, then, that there is a non-interesting Whole Number, there must be a "least" non-interesting Whole Number. However, since being the Least non-interesting Whole Number would be very interesting, indeed, we must conclude that there are no non-interesting Whole Numbers.
Finally, since there are plenty of Whole Numbers, and none of them is non-intereting, we must conclude that all Whole Numbers are interesting. QEF
That is not a very useful proof, but the style and conclusions are quite appropriate. Of course, the definition of "interesting" is not real clear - or should I say it is not Wholly clear.