Algebra Question regarding adjacent transposition

[sunnyday11]

Homework Statement

Prove that any transposition is a product of an odd number of adjacent transposition.

Homework Equations

The Attempt at a Solution

Let x=(i,j) Define gap(x) = j-i
By induction on gap:
If gap(x)=1 then already adjacent.
Suppose k= j-i
(i,j)=(i,i+1)(i+1,j)(i,i+1)
gap(i+1,j)=k-1
...

I don't really know how to complete the proof.
The gap is defined as the modulus of the value j-i in all cases above.

Thank you very much!

 

[micromass]

So gap(i+1,j)=k-1. What does the induction hypothesis tell you?

 

[sunnyday11]

By induction k-1 is odd.
So, gap(i,j)=(i,i+1) (odd number of adjacent transpositions) (i,i+1) = 1+ odd# + 1 = odd #

Thank you so much, but do you know how can I put this in a more formal presentation?

 

[micromass]

I think your proof is already quite formal. I don't see much way to improve it...