阅读短文，判断正误。

Dear Joe,

How are you? My sister and I are having our holiday. Yesterday we went to the library. We got there by bus. We read storybooks and had lunch there. The food was delicious. It was cheap, too. We went shopping in the afternoon. I bought a pair of shoes. I was very tired. We are planning to learn skating and do many other things during the holiday. We are sure we'll have a good time.

How about your holiday? Please write to me soon.

Miss you.

Alice

设函数$f\left(x\right)=\left\{\begin{array}{c}\left(a-2\right)x\left(x\ge 2\right)\\ {\left(\frac{1}{2}\right)}^x-1\left(x<2\right)\end{array}\right.$是R上的单调递减函数，则实数a的取值范围为(   )