Sequence of positive real numbers $(a_k)$ s.t. $lim_{k to infty} a_k = 1$ and $lim_{n to infty} a_1 a_2 dots a_n = 0$

I am looking for an example of a sequence of positive real numbers $(a_k)$ with $\lim_{k \to \infty} a_k = 1$ such that the sequence $(p_n)$ defined as $p_n=a_1 a_2 \dots a_n$ has limit 0 as $n \to \infty$.

Can anyone provide me with a concrete example, or maybe some hint or useful property of such a sequence?

Answer

Let consider

$$a_k=\frac{k}{k+1}$$

then

• $a_k \to 1$


• $\prod a_i =\frac12\frac23...\frac{k-1}k\frac{k}{k+1}=\frac1{k+1}\to 0$