Proving $e=2 \sqrt{\frac{4}{3}} \sqrt[4]{\frac{6\ 8}{5\ 7}} \sqrt[8]{\frac{10\ 12\ 14\ 16}{9\ 11\ 13\ 15}}...$

I've found an interesting infinite product formula without a proof: $$e=2 \sqrt{\frac{4}{3}} \sqrt[4]{\frac{6\ 8}{5\ 7}} \sqrt[8]{\frac{10\ 12\ 14\ 16}{9\ 11\ 13\ 15}}...$$ How can we prove it? Fow now I have no idea and would like you to help. Thanks!

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