Okay, this might be a little bit stupid question...
I'm studying ross elementary analysis.
And I'm trying to get some skills in writing formal proofs, instead of intuitive ideas.
I want to proof that the sequence (-1)^n = 1 or -1.
But how do I give a real proof for this ?
Okay, I'll try:
If n is even,
than ∃k∈N : n = 2k,
than (-1)^n = (-1)^2k=[(-1)^2]^k = 1^k
If n is uneven,
than ∃k∈N : n=2k-1,
than (-1)^n = (-1)^[2k-1] = -1^k,
Induction (prove that 1^k =1)
if 1^k = 1, than 1^(k+1)=1^k * 1 = 1^k = 1
By induction, 1^k =1 for all natural numbers.