I want to say "goodnight, see you later" in korean
So is this correct? 잘자요 나중에 봐요
Please could you explain why it's either wrong or correct and add eventual corrections?

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If you want to say"goodnight",you can use 안녕히 주무세요! If the person is your friend,you can use "잘 자". 
I think "see you later" maybe means  goodbye in Korean. So "내일 보자"is better~ 
I hope you like my answers~^^

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잘자요 . 나중에봐요 is fine!