A Cultural Question - Communicating with customers

Hello!

Can I ask a cultural question?

I sent an important file to an American customer, but, at first, he did not reply.

So the next day I sent another email, just checking if everything was fine.

He replied to the 2nd email, saying that yes, everything was fine, and thanking for my concern (not sure if "concern" is appropriate here; he just expressed gratitude for my 2nd email, "thanks for checking").

Should I reply to his answer? Something like: "I'm glad everything is fine. Thanks for letting me know. At any time you want, please do not hesitate to contact us. All the best, X". That's the situation:

1st email - me (file sent)

2nd email - me (checking politely)

3rd email - customer (everything is fine; thanks)

4th email (or not) - me (thanks for answering)

Would that be annoying, or polite? What about you, would you like to receive this reply to your somewhat late reply, or not, you would prefer to just end the conversation with your reply?

Thank you!!