A client finished her initial package of sessions, then emailed that she will not continue.
There are so many things that we don’t know in this scenario. But as you are already at this crossroad with this client, the most that I would suggest is to email her with a thank you of your own. Express that you enjoyed training with her. Leave the door open for her to contact you whenever she might want to in the future. If you have a newsletter, you could ask if she would like to receive it in the future.
It may just be that she feels upto exercising on her own. Do you feel she still needs your assistance? If so, tell her this in the email and why. She may have had a change of finances. It can be distracting to dwell on the possibilities. The only other way to find out her reason for discontinuing is to ask her directly. Whatever you decide to do, keep it to the point and professional.