I am trying to understand the proof of this problem, the link to the solution is given below. The proof uses two induction arguments. My question is why the first induction argument is necessary. The first induction argument shows: $$ ∀m ∈ N, \ \:exp^m(x) > 1 $$ The solution uses this to show that
$$ exp^m(x) < exp(exp^{m+d}) $$
for some positive integer 'd'.
Solution: (Problem #7) http://www.martinwernerlicht.de/math109/homework6-solutions.pdf
Problem: $$ \ For\:any\:x ∈ R\:we\:define\:repeated\:exponentiation\:as\:follows \\exp(x) = x, \:\: exp^{n+1}(x) = exp\:(\:exp^n(x)\:) \ \\ Prove\:the\:following\:statement\\ ∀x ∈ R : ∀m, n ∈ N_0 : (x > 1 ∧ m < n) → (exp^m(x) < exp^n(x)) \ $$
The first induction argument is in fact not necessary. As far as I can tell, the second argument is using it as justification for the claim that $a<\exp(a)$ (which is used a couple times, first with $a=\exp^m(x)$ in the base case and then with $a=\exp^{m+d}(x)$ in the induction step). But $a<\exp(a)$ is true for all $a\in\mathbb{R}$, not just for $a>1$, so there is no need to first prove that $\exp^m(x)>1$ for these steps.
(I suppose if you have already proved that $a<\exp(a)$ for $a>1$ but have not proved it for $a\leq 1$, then the first induction argument would have some role, but could equally well be replaced with just a proof that $a<\exp(a)$ for all $a$.)