Estimating the value of cosine with its power series

In Amann's Analysis, the following is stated:

Here $\mathbb{R}^{\times}$ denotes $\mathbb{R} \setminus \{ 0 \}$.

Corollary $II.7.9$ is the following

Here $s_n$ denotes the $n$th partial sum and $s$ the value of the series.

I have trouble seeing how corollary 7.9 shows the inequalities$$1-\frac{t^2}{2} < \cos t < 1- \frac{t^2}{2} + \frac{t^4}{24}.$$ Corollary 7.9 is about approximating the value of an alternating and decreasing series, but the absolute value of cosine series is not necessarily decreasing if $t>1$.