Exponential function proof problem

[rustyjoker]

Homework Statement

Big problem with exponential function proof assignment, need some help.

let
x≥0 and for every k$\in N$ there is $n_{k}$$\in N$ and

$x_{k1}$≥...≥$x_{k_{nk}}$ and $x_{k1}$+...+$x_{k_{nk}}$=x.
Proof: if $lim_{k→}∞ x_{k1}$=0 then $lim_{k→}∞<br />$ (1+$x_{k1}$)·...·(1+$x_{k_{nk}}$)=exp(x)=$e^{x}$

 

[Ray Vickson]

Homework Statement

Big problem with exponential function proof assignment, need some help.

let
x≥0 and for every k$\in N$ there is $n_{k}$$\in N$ and

$x_{k1}$≥...≥$x_{k_{nk}}$ and $x_{k1}$+...+$x_{k_{nk}}$=x.
Proof: if $lim_{k→}∞ x_{k1}$=0 then $lim_{k→}∞<br />$ (1+$x_{k1}$)·...·(1+$x_{k_{nk}}$)=exp(x)=$e^{x}$

There must be something wrong with the statement of hypotheses, because it allows me to take $x_1 = x,\: x_2 = x_3 = \cdots = x_n = 0,$ giving $\lim_{n \rightarrow \infty} (1+x_1) \cdot (1+x_2) \cdots (1+x_n) = 1+x.$

RGV

 

[rustyjoker]

Well, for an example if you think about the product
(1+0,009)(1+0,008)⋅...⋅(1+0,001)=1,045879514 and
exp(0,009+...+0,001)=1,046...
I think the idea is to proof that first $lim_{k→}∞$ $n_{k}=∞, then <br />$ $lim_{k→}∞$ $x_{k1}$=...=$lim_{k→}∞$ $x_{k_{nk}}$=x/$n_{k}$= 0. So we'd have
$lim_{k→∞}$ (1+x/$n_{k}$)$^{n_{k}}$ = $e^{x}$