No. "Daydream Believer" was written by John Stewart, and was a hit for the Monkees in 1967. (Producer Chip Douglas introduced them to the song; Peter Tork came up with the piano intro.)
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No, John Lennon wrote and recorded "Imagine" in 1971, about a year after The Beatles broke up.
No, Neil Diamond never recorded Daydream Believer. You may be think of the other Monkees hit I'm a Believer, which Neil Diamond wrote.
The Beatles sang Theodorakis in 1963, the title of the soon was "Honeymoon Song"
john Lennon from The Beatles (the beatles are the best band in history) John Lennon
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No. The song was written and recorded by Donovan, who had earlier helped the Beatles with "Yellow Submarine". There were stories that Paul McCartney added the "quite rightly" part in the song, but he didn't. There are also unconfirmed stories that one or more of the Beatles sang in the chorus of "Mellow Yellow".
John and George share most of "You Really Got a Hold On Me", but Paul joins in later.