Proving that the limit of a sequence is $> 0$ Let $u$ be the complex sequence defined as follows :
$u_0=i$ and
$ \forall n \in \mathbb N, u_{n+1}=u_n + \frac {n+1-u_n}{|n+1-u_n|}   $ .
Consider $w_n$ defined by $\forall n \in \mathbb N,w_n=|u_n-n|$ .
I have to prove that $w_n$ has a limit $> 0$.
Here is what I've proved so far:


*

*The sequence $Im(u_n)$ is decreasing and bounded by $0$ and $1$ hence convergent with a limit $\in [0;1]$

*$w_n$ is decreasing and bounded by $0$, hence convergent with limit $l \geq 0$


So all I need to prove now is that $l \neq 0$
I tried a proof with contradiction, but could not complete it...
Thanks for you help.

Well the other proofs are quite long...
I guess this is shorter.
A simple computation proves that the sequence $n-Re(u_n)$ is increasing.
Now by contradiction, if $l=0$ then $|Re(u_n-n)|=n-Re(u_n)$ converges to $0$
But $n-Re(u_n)$ is increasing.
Hence $\forall n \in \mathbb N, n-Re(u_n) = 0$
That's a contradiction.
I'd like to thank the users who gave a valid solution with a lower bound for $w_n$.
