# Verifying uniqueness of my tetration

Previous two posts:

Update: The first link only verifies continuity on $$\mathbb R$$, and so continuity cannot be used for complex heights. The limit of $${}^{z+n}a$$ as $$n\to\infty$$ converges to the same value for all $$z\in\mathbb C$$ since $$[\ln({}^\infty a)]^{z+n}\to0$$.

Let $$D_a$$ be the points where the below definition converges.

I believe I have managed to prove that for $$a\in(1,e^{1/e})$$ and $$z\in D_a$$,

$${}^za=\lim_{n\to\infty}\log_a^{\circ n}({}^\infty a-({}^\infty a-{}^na)[\ln({}^\infty a)]^z)$$

is the unique tetration under the conditions that

1. $${}^0a=1$$

2. $$\displaystyle{}^{z+1}a=a^{({}^za)}$$ for all $$z\in D_a$$

3. $$\displaystyle\lim_{n\to\infty}\frac{{}^\infty a-{}^{n+z}a}{{}^\infty a-{}^na}=[\ln({}^\infty a)]^z$$ as a limit over $$n\in\mathbb N$$ for all $$z\in D_a$$.

Here is my attempted proof:

Let $$b\pm\epsilon$$ refer to a value within $$\epsilon$$ of $$b$$ for simplicity.

From $$(3)$$ we know that for all $$\epsilon>0$$, there exists $$N$$ s.t. for all $$n>N$$,

$$\frac{{}^\infty a-{}^{n+z}a}{{}^\infty a-{}^na}=[\ln({}^\infty a)]^z\pm\epsilon$$

By continuity of exponentiation, this can be rewritten in terms of another $$\bar\epsilon>0$$:

$$\frac{{}^\infty a-{}^{n+z}a}{{}^\infty a-{}^na}=[\ln({}^\infty a)]^{z\pm\bar\epsilon}$$

Solving for $${}^{n+z}a$$ gives

$${}^{n+z}a={}^\infty a-({}^\infty a-{}^na)([\ln({}^\infty a)]^{z\pm\bar\epsilon})$$

Logging $$n$$ times and applying $$(2)$$ gives

$${}^za=\log_a^{\circ n}({}^\infty a-({}^\infty a-{}^na)([\ln({}^\infty a)]^{z\pm\bar\epsilon}))$$

Taking the limit as $$n\to\infty$$ gives us

$${}^za=\lim_{n\to\infty}\log_a^{\circ n}({}^\infty a-({}^\infty a-{}^na)([\ln({}^\infty a)]^{z\pm\bar\epsilon}))$$

However, from the first link, we know that this limit converges to a continuous function on $$D_a$$. So for this to be true for all $$\bar\epsilon>0$$, we must have

$${}^za=\lim_{n\to\infty}\log_a^{\circ n}({}^\infty a-({}^\infty a-{}^na)([\ln({}^\infty a)]^z))$$

Note also that the definitions of $${}^\infty a$$ and $${}^na$$ for natural $$n$$ in the above are defined by $$(1)$$ and $$(2)$$.