How can i show $\prod a_i<1$? Suppose we have $a_1&gt;1$ and for $i&gt;1,0&lt;a_i&lt;1$ how can i show that the product of infinite numbers less than 1 and $a_1$ is less than 1?
$$\prod_{i=1}^\infty a_i&lt;1.$$
Let $p_n=42+\frac1n$, $a_1=p_1&gt;1$, and $a_n=\frac{p_n}{p_{n-1}}&lt;1$ for $n&gt;1$. Then $$\prod_{i=1}^\infty a_i=\lim_{n\to\infty}p_n=42.$$
Well, $$\prod_{i=1}^\infty a_i= a_1 \prod_{i=2}^\infty a_i$$ so you want to prove that 
$\prod_{i=1}^\infty b_i =0$ if $0&lt;b_i&lt;1$. Which is false.

Taking logarithms, that would be equivalent to 
$ \sum_{i=1}^\infty \log b_i  = \sum_{i=1}^\infty c_i = - \infty$ if $c_i&lt;0$. 
Or $\sum_{i=1}^\infty d_i = + \infty$ if $d_i&gt;0$

If you know about convergent series, you know this is false; for example take $d_i=2^{-i}$