Re: limsup problem

On 2005-05-31, Saem T. <saemtuarg@xxxxxxxx> wrote:
> May you help me in the following problem, please? -Don't care about
> what is z_n, what is U_n and so on, I am sure the problem I'll show you
> is a very general one.
>
> --
> For any z_n in U_n there is a |k| beginning with which we have 1/|k|
> log |c_k(z_n)| leq log (1/ rho), i.e.,
>
> limsup_{|k|->infty} 1/|k| log |c_k(z_n)| leq log(1/rho).
> --
>
> leq stands for less or equal.
>
> The problem is: I do not understand the reason why
>
> ``For any z_n in U_n there is a |k| beginning with which we have 1/|k|
> log |c_k(z_n)| leq log (1/ rho)''
>
> is equivalent to
>
> limsup_{|k|->infty} 1/|k| log |c_k(z_n)| leq log(1/rho).

Not equivalent to. Implies.

If there exists a J for which c_j <= 5 for all j>=J then
limsup(c_j)<=5.

One the other hand, if c_j= 5+1/j, then limsup(c_j)<=5 too
(the limit exists and is five).
.

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