Proving what date an invoice was sent

Hi there,

I emailed an invoice to a customer 30 days ago. I chased a week later and they said they hadn't had it so I sent it again. They said they didn't have it again. I sent it to myself (worked fine so don't know what the issue is, their email is correct, maybe in their spam?) downloaded the pdf and emailed from my email address rather than send from the app. They're now trying to pay 30 days form the date they received it, rather than the invoice date. How do I show them proof that I sent it to the correct address direct through the app on the original date?

Many thanks