I'm an undergraduate math major (just completed my second year), and I had applied for a summer school (online due to the pandemic) in one of my interest areas. I have taken two courses in this area with the same professor (one of them was a graduate course if it matters). I did pretty well in both courses, i.e. got As.

I requested the aforementioned professor to write me a recommendation letter for the summer school, which they gladly did - but unfortunately, I did not get accepted (the school has about a 14% selection rate).

Should I inform the professor (i.e. my letter writer for this program) that I have been rejected? Do they already know, in which case I don't have to do anything? Do programs typically inform about acceptance/rejections to letter writers too, or do they only inform the applicants? This is my first time applying to a summer school, so I don't really know how this works.

Does the following email sound fine? Please suggest edits as you feel necessary:

Dear Prof. X,

Thank you for your help, I really appreciate it. Unfortunately, I wasn't accepted.

Best,
Y