Let [imath](a_n)_n[/imath] be a sequence such that for all [imath]n \in \mathbb{N}[/imath] it is [imath] 0<a_n<1[/imath] and [imath]a_n(1-a_{n+1}) >\frac{1}{4}[/imath].
Prove that [imath](a_n)_n[/imath] converges and find its limit.
I've tried this: since [imath] 0<a_n<1[/imath] for all [imath]n \in \mathbb{N}[/imath] it is [imath]a_n(1-a_{n+1}) >\frac{1}{4} \iff a_n > \frac{1}{4(1-a_{n+1})}[/imath], and it is [imath] a_{n+1} \ge \frac{1}{4(1-a_{n+1})} \iff (2a_{n+1}-1)^2 \le 0 \iff n \in \mathbb{N}[/imath]. So it is [imath]a_{n+1} \leq \frac{1}{4(1-a_{n+1})} < a_n \implies a_{n+1} < a_n[/imath] and so the sequence is decreasing.
Since by hypothesis the sequence is bounded below, it has a real limit [imath]l[/imath]: from [imath]a_n(1-a_{n+1})>\frac{1}{4}[/imath] taking the limit in the inequality if follows that it is
[math]l(1-l) \ge \frac{1}{4} \iff \left(l-\frac{1}{2}\right)^2 \leq 0 \iff l=\frac{1}{2}[/math]I am not sure about this because I'm not sure if my proof of the decreasing sequence is correct (I think it is because I've proved that [imath]a_{n+1}[/imath] is less than something that is less than [imath]a_n[/imath], could this work?) and I'm not sure when I take the limit in the inequality, because I have an estimation of the possible limit [imath]l[/imath] and not an equality (but I think that this works because the estimation leads to the fact that [imath]l[/imath] can only be [imath]\frac{1}{2}[/imath] or the other true sentences I've written wouldn't hold; is this correct?). Thank you.
Prove that [imath](a_n)_n[/imath] converges and find its limit.
I've tried this: since [imath] 0<a_n<1[/imath] for all [imath]n \in \mathbb{N}[/imath] it is [imath]a_n(1-a_{n+1}) >\frac{1}{4} \iff a_n > \frac{1}{4(1-a_{n+1})}[/imath], and it is [imath] a_{n+1} \ge \frac{1}{4(1-a_{n+1})} \iff (2a_{n+1}-1)^2 \le 0 \iff n \in \mathbb{N}[/imath]. So it is [imath]a_{n+1} \leq \frac{1}{4(1-a_{n+1})} < a_n \implies a_{n+1} < a_n[/imath] and so the sequence is decreasing.
Since by hypothesis the sequence is bounded below, it has a real limit [imath]l[/imath]: from [imath]a_n(1-a_{n+1})>\frac{1}{4}[/imath] taking the limit in the inequality if follows that it is
[math]l(1-l) \ge \frac{1}{4} \iff \left(l-\frac{1}{2}\right)^2 \leq 0 \iff l=\frac{1}{2}[/math]I am not sure about this because I'm not sure if my proof of the decreasing sequence is correct (I think it is because I've proved that [imath]a_{n+1}[/imath] is less than something that is less than [imath]a_n[/imath], could this work?) and I'm not sure when I take the limit in the inequality, because I have an estimation of the possible limit [imath]l[/imath] and not an equality (but I think that this works because the estimation leads to the fact that [imath]l[/imath] can only be [imath]\frac{1}{2}[/imath] or the other true sentences I've written wouldn't hold; is this correct?). Thank you.