\(B=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}+.....+\frac{1}{5^{2018}}+\frac{1}{5^{2019}}\)
\(\Rightarrow5B=1+\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+.......+\frac{1}{5^{2017}}+\frac{1}{5^{2018}}\)
\(\Rightarrow5B-B=1-\frac{1}{5^{2019}}\)
\(\Rightarrow4B=1-\frac{1}{5^{2019}}\)
\(\Rightarrow B=\frac{1-\frac{1}{5^{2019}}}{4}< \frac{1}{4}\left(đpcm\right)\)