But I saw a lot of people wrote "I play tennis everyday".
Is "I play tennis every day" the right way?
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You are right, everyday is an adjective and it should not be used in this way.
Native English speakers learn most of their vocabulary by listening. When they write things out, they sometimes make mistakes with the spelling of words that sound the same: this is an example.
Here is some more information that clearly explains the difference between everyday and every day.