The activation email isn't a panacea for users fumble fingering (or misremembering, or not knowing) their email address. Users who don't receive it will either a) ignore it if you let them use the application anyway or b) frequently bounce hard if they don't get it, because they assume naturally that your Googles are broken.
The single most compelling reason to send people activation emails is -- I kid you not -- to remind them that they signed up for your website and how to get back to your website. A secondary consideration is not proving that they got their inbox right but proving that they didn't get someone else's inbox wrong.
The single most compelling reason to send people activation emails is -- I kid you not -- to remind them that they signed up for your website and how to get back to your website. A secondary consideration is not proving that they got their inbox right but proving that they didn't get someone else's inbox wrong.