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$\begingroup$Are you sure you're not missing an assumption? Under the appropriate additional assumption(s), a few seconds with Google will reveal many proofs on the internet, including on this site.$\endgroup$
$\begingroup$Please follow @gung's advice and let us know whether this is a self-study question. It's somewhat unclear what you are asking at the moment - are you asking what conditions are needed for the sample mean and variance to be independent, or are you asking for a proof that they are independent, but did not realise this required additional assumptions to hold?$\endgroup$
[self-study]
tag & read its wiki. $\endgroup$