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Deleting email in Gmail is easy. I cannot say the same thing about Exchange support email, like Live@edu or Office 365.
You seem to need to do it in stages, as the web client can only handle so many emails at a time. Because Live@edu and Office 365 data is on Exchange, it doesn't delete it straight away. It holds it so you can recover it if necessary.
Select as many emails as you can (your email client will warn you when you have reached the limit), and right click and then hit Delete.
To read more on this gallery, visit the iGeneration column, where you can read more about migrating your email to Hotmail or Gmail.