Using the result proved for any two functions and induction, prove that for all n in N, if ...?

Question

f_i:X_i -> X_i+1, i= 1,2,........n are all invertible functions, then the composition f_n o f_n-1 o........o f_1 is also invertible and (f_n o f_n-1 o........o f_1)^-1 = (f_n)^-1 o.......o (f_2)^-1 o (f_1)^-1 .

HINT: Use induction

Answer 1

That's pretty trivial, actually.

Check it directly for the case where n = 2. Then for the inductive step apply the k = 2 case to two functions. One is the composition of n-1 functions, which you know how to invert by the inductive assumption. The other is the nth function.