It really depends on definitions.
There are two ways to "compactify" the real line. One is to add two points, $+\infty$ and $-\infty$. The other way is to add one point at infinity, $\infty$. The latter is, in a sense, more canonical - every space has a "one-point" compactification, while only and "ordered space" has a two-point compactification.
Alternatively, you might say that $x_n\to\infty$ if $|x_n|\to +\infty$.
I really dislike the usage of $\infty$ to mean $+\infty$ for this reason, so I agree with the book's usage. It might be confusing that $+\infty$ is different from $\infty$, but infinity is confusing.