Let $\{\alpha_n\},\{\beta_n\}$ be real valued sequences, such that $\alpha_n<\beta_n$, $\alpha_{n-1}<\beta_n$ and $\beta_1>0$, such that $\alpha_n\to\alpha$, $\beta_n\to \beta$ where $\alpha\le\beta$.
How to be sure that such $\{\alpha_n\},\{\beta_n\}$ exist for each $-\infty\le\alpha\le\beta\le+\infty$?
My earlier comments ended up a bit messier than I would like. Hence I decided to write up an answer and be a bit more careful about the details:
I assume that $- \infty < \alpha \le \beta < \infty$ and leave the remaining cases up to you. (They are simpler but rather annoying, since there are several cases to consider.)
For each $n$ let $$ \alpha_n = \alpha - \frac{1}{n} $$ and $$ \beta_n = \beta + \frac{2 | \beta | +1 }{n}. $$ This guarantees $\alpha_n \to_{n \to \infty} \alpha$ and $\beta_n \to_{n \to \infty} \beta$.
Moreover, since $\alpha \le \beta$ and $\alpha_n < \alpha \le \beta < \beta_m$ for all $m,n \in \mathbb N$ we have that $\alpha_n, \alpha_{n-1} < \beta_n$.
Finally $\beta_1 = \beta + 2 | \beta | + 1 > 0$.