Quick notation+statement verification

[bomba923]

Do you agree that, $$\forall k \in \left[ {a,b} \right]\;{\text{where}}\;\left( {a,b,k} \right) \in \mathbb{Q}^3$$,
$$\exists \,\varepsilon > 0{\text{ such that}}\;\forall n \in \mathbb{N},\;\left( {\left\{ {k_1 ,k_2 , \ldots ,k_n } \right\} - a} \right) \subseteq \varepsilon \left\{ {0,1,2, \ldots ,\left\lfloor {\frac{{b - a}}{\varepsilon }} \right\rfloor } \right\}$$

|*Is this True or False ?

 

[matt grime]

i don't think it makes sense until you say what the k_n are. don't bother with the symbols just write it in english.

 

[bomba923]

Sorry; the whole mess seems to simplify down to this statement:

$$\forall \left\{ {k_1 ,k_2 , \ldots ,k_n } \right\} \subset \mathbb{Q}\;{\text{where }}k_1 < k_2 < \ldots < k_n ,$$
$$\exists \,\varepsilon > 0\;{\text{such that}}\;\forall n \in \mathbb{N},\;\left\{ {k_1 ,k_2 , \ldots ,k_n } \right\} \subseteq \varepsilon \left\{ {0,1,2, \ldots ,\left\lfloor {\frac{{k_n - k_1 }} {\varepsilon }} \right\rfloor } \right\}$$

*|is this True or False?