Solving Symmetric Group Induction Proof: Hints for Double Induction

[Bashyboy]

Homework Statement

Consider the symmetric group $S_n$. I am trying to establish that $(i,i+1)=(1,2,...,n)(i-1,i)(1,2,...,n)^{-1}$

Homework Equations

The Attempt at a Solution

I am trying to decide whether I need double induction or not. I have done several calculations to see whether I can get away with one induction, but it isn't clear to me whether this is possible. I could use some hints.

 

[fresh_42]

Why don't you simply apply the mappings for the cases $j< i-2\, , \,j > i-1$ and $j \in \{i-2,i-1\}$ if read from left to right? It is no recursive definition but an explicit formula, so why use induction?

 

[Bashyboy]

Hold on! I forgot that I proved that $\sigma (x_1,x_2,...,x_n) \sigma^{-1} = (\sigma(x_1),...,\sigma(x_n))$, which makes the problem trivial.