content_image/300034.jpg content_image/300033.jpg content_image/300035.jpg content_image/300032.jpg content_image/300031.jpg content_image/300030.jpg content_image/300029.jpg content_image/300028.jpg
In this case, for each n ∈ N , there exist integers k n 0 between 1 and N such that 

Therefore, it can be verified that 

The last inequality implies the existence of numbers η$_{n}$ ∈ [ − k n $_{0}$, k n $_{0}$] satisfying 

<img src='content_image/300034.jpg'>

<img src='content_image/300033.jpg'>

<img src='content_image/300035.jpg'>

Suppose that g ( s ) = ( g$_{1}$ ( s ) , g$_{2}$ ( s ) , . . . , g$_{m}$ ( s )) , where each g$_{i}$, 1 ≤ i ≤ m, is a real-valued function. In accordance with (4.6), for each n ∈ N , there is an integer j$_{n}$, 1 ≤ j$_{n}$ ≤ m, with 

<img src='content_image/300032.jpg'>

where τ$_{n}$ = τ$_{n}$ + η$_{n}$, n ∈ N . Since the function g is uniformly continuous, there exists a positive number Δ ≤ 1 , which does not depend on the sequences { t$_{n}$ } and { τ$_{n}$ } , such that both of the inequalities 

and 

are valid for s ∈ [ − Δ , Δ] . Thus, we have for s ∈ [ − Δ , Δ] that 

<img src='content_image/300031.jpg'>

<img src='content_image/300030.jpg'>

<img src='content_image/300029.jpg'>

For each n ∈ N , one can find numbers s n $_{1}$, s n $_{2}$, . . . , s n m ∈ [ − Δ , Δ] such that 

<img src='content_image/300028.jpg'>

