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I edited the question to say series instead of sequence, since I think that's what you meant (if these were terms in a series, then since they're all positive the series is increasing, which would be the whole proof).
Anyway, if we label the terms of the sequence as `s_n=(4n)/(n+3),` then if we can show that `s_(n+1)-s_n>0` for any `n,` this proves by definition that our sequence is increasing for all `n` .
`=12/((n+4)(n+3))>0,` (since `n>=0`)
as we wished to show. This finishes the proof. The third sentence in the link describes an alternate method that works for this example.
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