Is this compositions of sequences bounded?

Question

Is the following statement true? Let $m^{1},m^{2}:\mathbb{N}\to\mathbb{N}$ be sequences s.t. $m^{i}_{n}\leq m^{i}_{n+1}$ and $\lim_{n\to\infty}m^{i}_{n}=\infty$. Let $a:\mathbb{N}\to \mathbb{R}_{>0}$ be a sequence s.t. $a_{n+1}<a_{n}$ and $\lim_{n\to\infty}a_{n}=0$. Suppose the sequences $\frac{a_{n}}{a_{m^{i}_{n}}}$ are bounded then $\frac{a_{n}}{a_{m^{1}_{m^{2}_{n}}}}$ is bounded as well.

Answer 1

Yes. You can write

$$\frac{a_n}{a_{m^1_{m^2_n}}}=\frac{a_n}{a_{m^2_n}}\cdot\frac{a_{m^2_n}}{a_{m^1_{m^2_n}}}$$ where the first quotient is bounded by your assumptions, and the second quotient is a subsequence of $\frac{a_n}{a_{m^1_n}}$, hence bounded.