Old Email

I had a non mac email account which I had set up in Mail and it was downloading mail normally to my mac. Then I got a mac email account, set it up in mail and began receiving mail on both accounts. Then I closed the non mac account and once it was completely deactivated at the service provider I noticed that it was continually trying to download mail from this source and showing it as offline. To remedy this I went into Mail preferences, account and deleted it. I was expecting all of the several years worth of old mail to still be available in its folder. However it does not appear anywhere. Where would it be and how can I get it to show up in Mail when the account is no longer active? Thank you in advance for your assistance!