How to prove that an email has been sent?
Solution 1:
Most people are shocked to hear this, but email is not actually guaranteed to ever reach the intended destination.
It might fail for a variety of reasons. It could simply fail to arrive (for several reasons) or could be getting redirected to the recipients spam folder for some reason. If you think an email hasn't been delivered, you should send it again. I usually forward the original so that recipient knows I've been trying.
You can turn on “read receipt” or “return receipt”, which will send an email back to you automatically, but this feature is often disabled by recipients because it has been so often abused by spammers in the past.
Will covered just about everything else in his answer.
Solution 2:
There is nothing 100% that you can use to prove on your own.
If you had your own mail server, you can show outgoing logs - however, they can be falsified.
You can use read receipts, but again, they can be falsified.
The best bet is if you use a third party mail server that is impartial, and they can provide logs, that should stand as proof.
You may have luck if you email Yahoo (and say you are willing to pay for their time on the matter) and try to get a log / proof that an email was sent on a time/date.
I am not sure you can force them in to this unless the place you need it for are involved in a criminal matter... It may work for civil, but you would need a court order.
Failing this, if it is in your outbox with a time and date, this is hard/impossible to fake on a web based email server (just check that changing your date/time and sending doesn't fake this). Then, print out and if you need to go to court, you could always have a laptop, 3g stick and projector and prove that it was sent... but again, only if it can be prooved that you can't fake this.
Solution 3:
Unfortunately, as with physical letters, proof of sending is not proof of receipt.
While the e-mail shows as sent in your mail client and it will probably be shown as sent in your ISPs mail server, there will be (many) other mail servers between there and the final recipient. At each one of these the e-mail could have got lost or delayed, so it could well be that your landlord's estate agent is telling the truth (though I'll admit it's more likely that they are lying).
To have proof of receipt you'll need the equivalent of registered post here in the UK where the recipient has to sign for the letter.