What is the Proper Notation for Inductive Proofs with Multiple Variables?

[jbusc]

Hi, I have to write several inductive proofs for a class.

Typically, 'n' is used to denote the iterative variable in the problem statement. Then I show the case for n = 1 (or however appropriate for the proof) then proceed to show that if valid for n = k, then valid for n = k+1

However, there are more variables now in the given problem statement (using variables n, k, m, etc) which leaves me uncertain as to how to properly label the inductive step variable. I feel re-using n, k, or m would create additional confusion, as it would if I used alternative variable labels that are not traditionally used to refer only to integers (a, b, c, x, y, z, etc)

How should I alleviate this? Am I clear enough? It's kind of hard to describe...

 

[d_leet]

I don't think it will really make a difference as long as it is clear from your proof how you are using the variable, but I guess you could use j, or l if you want to stick the the letters of the alphabet surrounding n, k, and m.

 

[0rthodontist]

You can also use n for the induction variable, or if not n, whatever variable you happen to be inducting on. It's unnecessary complication to use k in the first place. You just argue:

Assume S(n)
   ...
   S(n+1)
S(n) --> S(n+1) (conditional proof)
for all n, S(n) --> S(n+1) (universal generalization)

Since n is bound by a quantifier outside of the conditional proof, there is no scope conflict.