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` .

`s_(n+1)-s_n=(4(n+1))/((n+1)+3)-(4n)/(n+3)=(4n+4)/(n+4)-(4n)/(n+3)`

`=((4n+4)(n+3))/((n+4)(n+3))-(4n(n+4))/((n+4)(n+3))=(4n^2+16n+12)/((n+4)(n+3))-(4n^2+16n)/((n+4)(n+3))`

`=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.