Hermite Functions (show hermite function belongs in schwartz class )

[jac7]

I have been given this question (in the attachment).

I have a deifnition for what it means for a function to be in the schwartz class, but I don't know how to start showing that the hermite function belongs to it?
I have attempted to write out the first couple of terms using the n+1 formula for the hermite functions but i ended up getting 0 for n=2 term!

I also know, using the hint, when n=0, what the Fourier transform of the hermite function at n=0 is.

If someone could please give me some guidence, it would be a great help!

 

[jac7]

sorry here is the attachment