Re: Problem with limiits

• To: mathgroup at smc.vnet.net
• Subject: [mg65845] Re: Problem with limiits
• From: Roger Bagula <rlbagulatftn at yahoo.com>
• Date: Tue, 18 Apr 2006 06:56:40 -0400 (EDT)
• References: <200604160544.BAA07913@smc.vnet.net> <e1vdsq\$967\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Andrzej Kozlowski wrote:

>>
>
>
>
> For any sequence of real numbers a1,a2, a3 .... converging to a real
> number L,  any infinite subsequence of it will converge to the same
> limit L. This is a very basic fact and is in fact true in any
> complete metric space (a subsequence of a convergent sequence is a
> Cauchy sequence, and therefore, in a complete metric space, is itself
> convergent.)
>
> Given the above,  you only need the fact that there are infinitely
> many primes (proved by Euclid and before him by Eudoxus).
>
> Andrzej Kozlowski
>
What you are saying is that if I have a composite number c[m]
them the Limit:

Limit[(1+1/c[m])^c[m],m->Infinity]=E

as well.
Limit[(1-1/m)^m,m->Infinity] =1/E

Limit[(1-1/c[n])^c[n],n->Infinity]*Limit[(1+1/Prime[n])^Prime[n],N->Infinity]=1
and
Limit[(1-1/m)^m,m->Infinity]*Limit[(1+1/m)^m,m->Infinity]=1
This does give:
Limit[(1 - 1/m^2)^m, m -> Infinity]=1