(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let x ∈ N. Show that there exists for each n ∈ N a natural number denoted by x^n (this is just a notation, but should tell you what we are doing)such that x^1 =x and x^σ(n) =x·x^n.

2. Relevant equations

σ(n) = n + 1

3. The attempt at a solution

So far my answer is:

1. Let S = {n ∈ N | x^1 = x and x^σ(n) = x * x^n}.

We know that x^1 = x for any x ∈ N and σ(1) = 1+1 = 2. So x^σ(n) = x^2 = x * x^1. Therefore 1 ∈ S. Now assume n ∈ S. We know that x^1 = x regardless of n. Since σ(n) = n + 1, it follows that σ(σ(n)) = σ(n) + 1 = n + 1 + 1. So x^σ(n) = x^(n+1+1) = x^(n+1) * x^1 = x * x^σ(n). Therefore σ(n) ∈ S. By the principle of induction we see that S = N.

My question is: Do I need to go into more detail in showing x^1 = x for this to be a valid proof? and if so how can i do that?

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# Principle of induction problem

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