My intuition is that this sequence is not convergent. I started by proving with induction that the sequence is strictly decreasing:
Base case:
$n = 1, a_1 = 2$
$n = 2, a_2 = -3 – (2)^2 = -7 < a_1$
Induction step:
Assume $a_n < a_{n-1}$
$a_{n+1} = -3 – a_n^2 < -3 – a_{n-1}^2 < a_n$
Thus $(a_n)_{n=1}^\infty$ is strictly decreasing
Next I tried to arrive at a contradiction using the definition of the limit of a sequence
ATC that $a_n$ converges to some L $\in$ R
Let $\varepsilon$ > 0, N $\in$ N to be defined later such that $\forall$n $\geq$ N we have
| $a_n$ – L | < $\varepsilon$
L – $\varepsilon$ < $a_n$ < L + $\varepsilon$
At this point I'm unsure if this proof is going on the right direction or if I took the wrong approach
Best Answer
Assume that $a_n$ converges to some $L\in\mathbb{R}$. Then $L=-3-L^2$, but this equation hasn't real solutions.