If $|c|>2$ and $z_n=z_n-1^2+c$ with $z_0=0$ then $z_n rightarrow infty$

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Let $c in mathbbC$ and $f(z)=z^2+c$. Then define a sequence: $z_0=0$, $z_n=f(z_n-1)$ for all $n in mathbbN$. Show that $|c|>2 Rightarrow z_n rightarrow infty$. I had no clue which theorem I could use in this case in order to show this. I'd be thankful for any advice.




complex-analysis




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  • (Terrible title.) No need for a "theorem" here, just find a lower bound for $|z_n|$ in terms of $|z_n-1|$ and iterate.
    – Did
    Aug 22 at 11:26






  • 2




    Please refer to "escape criterion" here.
    – Ng Chung Tak
    Aug 22 at 18:25















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Let $c in mathbbC$ and $f(z)=z^2+c$. Then define a sequence: $z_0=0$, $z_n=f(z_n-1)$ for all $n in mathbbN$. Show that $|c|>2 Rightarrow z_n rightarrow infty$. I had no clue which theorem I could use in this case in order to show this. I'd be thankful for any advice.




complex-analysis




share|cite|improve this question






















  • (Terrible title.) No need for a "theorem" here, just find a lower bound for $|z_n|$ in terms of $|z_n-1|$ and iterate.
    – Did
    Aug 22 at 11:26






  • 2




    Please refer to "escape criterion" here.
    – Ng Chung Tak
    Aug 22 at 18:25













up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
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1





Let $c in mathbbC$ and $f(z)=z^2+c$. Then define a sequence: $z_0=0$, $z_n=f(z_n-1)$ for all $n in mathbbN$. Show that $|c|>2 Rightarrow z_n rightarrow infty$. I had no clue which theorem I could use in this case in order to show this. I'd be thankful for any advice.




complex-analysis




share|cite|improve this question














Let $c in mathbbC$ and $f(z)=z^2+c$. Then define a sequence: $z_0=0$, $z_n=f(z_n-1)$ for all $n in mathbbN$. Show that $|c|>2 Rightarrow z_n rightarrow infty$. I had no clue which theorem I could use in this case in order to show this. I'd be thankful for any advice.





complex-analysis





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 22 at 11:26









Did

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asked Aug 22 at 11:23









Thesinus

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  • (Terrible title.) No need for a "theorem" here, just find a lower bound for $|z_n|$ in terms of $|z_n-1|$ and iterate.
    – Did
    Aug 22 at 11:26






  • 2




    Please refer to "escape criterion" here.
    – Ng Chung Tak
    Aug 22 at 18:25

















  • (Terrible title.) No need for a "theorem" here, just find a lower bound for $|z_n|$ in terms of $|z_n-1|$ and iterate.
    – Did
    Aug 22 at 11:26






  • 2




    Please refer to "escape criterion" here.
    – Ng Chung Tak
    Aug 22 at 18:25
















(Terrible title.) No need for a "theorem" here, just find a lower bound for $|z_n|$ in terms of $|z_n-1|$ and iterate.
– Did
Aug 22 at 11:26




(Terrible title.) No need for a "theorem" here, just find a lower bound for $|z_n|$ in terms of $|z_n-1|$ and iterate.
– Did
Aug 22 at 11:26




2




2




Please refer to "escape criterion" here.
– Ng Chung Tak
Aug 22 at 18:25





Please refer to "escape criterion" here.
– Ng Chung Tak
Aug 22 at 18:25
















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