# Simple question on limits and ceiling function

Suppose that $$f : \mathbb{N} \to \mathbb{R}_{++}$$, $$g : \mathbb{N} \to \mathbb{R}_{++}$$ are such that $$F := \lim_{n \to \infty} f(n)$$ and $$G := \lim_{n \to \infty} g(x)$$ both exist. Let $$0 and let $$\lceil \cdot \rceil$$ denote the ceiling operator.

Claim. Suppose that

$$\lim_{n \to \infty} \frac{(f(n))^{\lceil an \rceil}}{(g(n))^{\lceil b n \rceil}} = c$$

where $$0.

Then the following hold:

1. By the density of $$\mathbb{Q}$$ in $$\mathbb{R}$$, there exist $$k,l such that $$\lim_{n \to \infty} k/n = a$$ and $$\lim_{n \to \infty} l/n = b$$. Therefore

$$\lim_{n \to \infty} \frac{(f(n))^{\lceil an \rceil}}{(g(n))^{\lceil b n \rceil}} = \lim_{n \to \infty} \left( \frac{(f(n))^a}{(g(n))^b} \right)^n$$

1. $$F^a = G^b$$ Why? Otherwise, the limit above would be either $$0$$ (when $$F^a) or positive infinity (when $$F^a>G^b$$).

All of these seem intuitively true to me. Is there anything I should be careful about or can I safely assume these two facts going forward?

I would reason like this: $$\frac{[an]}{[bn]}=\frac{an-O(1)}{bn-O(1)} \to a/b$$ so $$\frac{[an]}{[bn]}\log f(n)-\log g(n) \to a\log F/b-\log G$$; but $$\log c/[bn]\to 0$$ so the hypothesis implies (taking log) that $$\frac{[an]}{[bn]}\log f(n)-\log(n) \to 0$$ so $$a\log F=b\log G$$
Note that any $$a,b >0$$ will work too