Inequalities involved in the proof of transcendence of $[0,10,10^{2!}, \dots]$

Question

In proof of transcendence of the simple continued fraction $[0,a_1,a_2, \dots]$ in which $a_k=10^{k!}$ (Hardy, et al's Theory of Numbers) it uses the following two inequalities:

i. $(1+\frac{1}{10})(1+\frac{1}{10^2}) \dots (1+\frac{1}{10^n})<2$ (for all n) and

ii. $2 \times 10^{1!+2!+\dots+n!}<10^{2(n!)}$.

How the two mentioned inequalities holds?

Answer 1

The hint:

Use $$\ln(1+x)\leq x$$ By this way we can get that even $$\prod_{k=1}^n\left(1+\frac{1}{10^k}\right)<\sqrt[9]e.$$

The second inequality is obviously true by induction.

Good luck!