Infinite product

Prove that for any positive number $t$ the following holds -
$$\prod_{n=1}^{\infty}\left(\frac{1}{2}+\frac{1}{2}e^{2^{-n}t}\right)=\frac{e^t-1}{t}$$
 (solved, solution to be posted)
Interesting corollary appears when we change $t\to\log t$. We have $$\log t=(t-1)\left(\prod_{n=1}^{\infty}\left(\frac{1}{2}+\frac{1}{2}t^{2^{-n}}\right)\right)^{-1}$$

Recursive sequence

Let $x\in (0,1)$. Define sequence $a_1=x$ and $$a_{n+1}=a_n-a_n^2, \ \ \forall n\ge 1$$.
Prove that $$\lim_{n\to\infty} n a_n=1$$

A solution could be done along the lines of Cesaro-Stolz theorem.