About a week ago I answered the question Curious limit of a sequence used to prove Etemadi's SLLN where I proved the following elementary lemma of Etemadi:
Lemma (Etemadi). Let $\{x_{n}\}_{n=1}^{\infty}$ be a sequence of non-negative real numbers. If $$\lim_{n\to\infty} \frac{1}{\left\lceil{r^n}\right\rceil}\sum\limits_{i=1}^{\lceil{r^n}\rceil}x_{i} = c$$ for all $r >1$ with $r\in \mathbb{R}$, then $$\lim_{n\to\infty} \frac{1}{n}\sum\limits_{i=1}^{n}x_{i} = c.$$
I'm wondering, how far does Etemadi's lemma generalise? I'm looking for a "nontrivial" larger class of functions $f:\mathbb N \to \mathbb R$ for which the title statement holds true.
I will also accept as an Answer any sequence $x_n$ such that the limit condition of the lemma is satisfied, but not the conclusion, or more generally a function $f$ such that $f(\lceil r^n \rceil) \to L$ for every $r>1$ but $f(n)$ does not converge to $L$ as $n\to\infty$ along the integers.
PS: I don't expect this to be relevant in the solution, but this is superficially reminiscent of an infamous exercise that relies on the Baire Category theorem:
If $f$ is continuous and $f(r^n) \to L$ for all $r>1$, then $f(x) \to L$ as $x\to\infty$ (along the real numbers).
This is usually written using the function $F(t) = f(e^t)$, so that the assumption is instead "If $F$ is continuous and $F(nx) \to L$ for every $x>0$". This exercise has been on MSE before, see for instance this link (thanks PhoemueX!)
