Let $($an$)$ be the sequence where a$1=0\mathrm{.}1,$ a$2=0\mathrm{.}12,$ a$3=0\mathrm{.}123,$ a$4=0\mathrm{.}1234,$ and so on$.$ This pattern does not change when you reach double digits. For example, a$12=0\mathrm{.}123456789101112.$ Prove that $($an$)$ converges. $($Hint: Use the Monotone Convergence Theorem$)$