Real numbers — digits of a sequence approaching a terminating decimal

Let $\alpha=\sum_{\ell=1}^{2025}\dfrac{a_{\ell}}{10^{\ell}}$ with $a_{\ell}\in\{0,1,…,9\}$ for all $1\le\ell\le2025$ and $a_{2025}\neq0$. Let $(\beta_n)_{n\ge1}$ be a strictly increasing sequence of positive real numbers in $(0,1)$ that converges to $\alpha$. For each $n\ge1$, write $\beta_n=\sum_{\ell=1}^{\infty}\dfrac{b_{n,\ell}}{10^{\ell}}$ with $b_{n,\ell}\in\{0,1,…,9\}$, for the infinite decimal expansion of $\beta_n$. Which of the following statements are necessarily true?