Let $a \in (0,1)$ and a monotonically (strictly) decreasing sequence of non-negative numbers $t_n \to 0$ be given.
Define $$b_n = t_n(1+a) + a.$$
How can I show that for every $\epsilon >0$, there exists an $n_0$ such that $n \geq n_0$ implies $$b_1\times b_2 \times \cdots \times b_n \leq \epsilon?$$
Is it possible to quantify $n_0$ in terms of the $\epsilon$ and $a$?
To show this one could use the binomial theorem, but the cross terms are hard to make small.