I am stuck in Problem 1.3.18 in Berkeley problems in Mathematics. Without looking back in the section of solutions, I want to ask for a hint.
The problem is as follows: Let $\{b_{i}\}$ be positive real numbers with $\lim_{n\rightarrow \infty}b_{n}=\infty$ and $\lim_{n\rightarrow \infty}\frac{b_{n}}{b_{n+1}}=1$ Assume also $b_{1}< b_{2}
My thoughts are to translate this problem into $a_{i}>1$, $\lim a_{i}=1$, and prove $B_{m,n}=\prod^{m}_{n}a_{i}$ is dense in $(1,\infty)$. But this does not make the problem any easier: for example, given $1+\delta$ and $\epsilon$, how can I show there must be some $m,n$ such that $B_{m,n}$ is in the $\epsilon$ neighborhood of $1+\delta$? When $a_{i}$ approach 1, it could will ignore $1+\delta$ and 'jump' straightforward from $m\in \mathbb{Z}$ to $1+\frac{1}{100}\delta$, for example. And $m\left(1+\frac{1}{100}\delta\right),m,\left(1+\frac{1}{100}\delta\right)$ may all be totally out of $1+\delta$'s $\epsilon$ neighborhood. Similarly the $a_{i}$'s after $\left(1+\frac{\delta}{100}\right)$ may shrink so quickly that both $\prod^{\infty}_{L} a_{i}$ and $m\prod^{\infty}_{L} a_{i}$ are outside of the $\epsilon$ neighborhood as well(one is too small, the other is too large). In short I do not know how to prove constructively the $B_{m,n}$s must fall into every open subset in $(1,\infty)$.