Let $\{a_n\}$ be a sequence such that $a_1 = 1 , a_{n+1} = \sqrt{6+a_n}$. We want to prove that this sequence is monotonic (increasing) and it is also bounded. So we can prove that it is convergent. I don't know how to start proving it that it is bounded. It is very easy to see that it is convergent to 3 by testing some number or if we think that we know it is convergent and then finding its limit in infinity. But I don't know how I can prove that it is bounded and monotonic so I can prove that it is convergent.
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Also: https://math.stackexchange.com/q/850472/42969, https://math.stackexchange.com/q/1847377/42969, https://math.stackexchange.com/q/115501/42969 – Martin R Oct 27 '17 at 18:54
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Hint: show by induction that $a_n < 3$. – Robert Israel Oct 27 '17 at 18:57