show that, for n >= 1
(1+10^-1)(1+10^-2)...(1+10^-n) <=2
Induction is not suitable for proving this result. The best method is to take logs and use the well-known inequality $\displaystyle \ln(1+x)\leqslant x.$
This tells you that $\displaystyle \sum_{k=1}^n\ln(1+10^{-k})\leqslant\sum_{k=1}^n10^{-k} = {\textstyle\frac19}(1-10^{-n})<{\textstyle\frac19}.$ Therefore $\displaystyle \prod_{k=1}^n(1+10^{-k})<e^{1/9}<2.$