6 The Proof of Theorem 2.3

In this chapter we will establish Theorem 2.3 and hence immediately Theorem 2.2 follows [39].

6.1 Proof of Theorem 2.3 (part(2))

We begin with the proof of Theorem 2.3 (part(2)).

Proof: We argue by way of contradiction that the result does not hold. Then for each l greater-than-or-equal-to 1, we can find points y 1 Superscript left-parenthesis l right-parenthesis Baseline comma ellipsis comma y Subscript k Superscript left-parenthesis l right-parenthesis and z 1 Superscript left-parenthesis l right-parenthesis Baseline comma ellipsis comma z Subscript k Superscript left-parenthesis l right-parenthesis in double-struck upper R Superscript d satisfying (2.3) with delta equals 1 slash l but not satisfying (2.4). Without loss of generality, we may suppose that diam StartSet y 1 Superscript left-parenthesis l right-parenthesis Baseline comma ellipsis comma y Subscript k Superscript left-parenthesis l right-parenthesis Baseline EndSet equals 1 for each l and that y 1 Superscript left-parenthesis l right-parenthesis Baseline equals 0 and z Subscript l Superscript left-parenthesis 1 right-parenthesis Baseline equals 0 for each l. Thus for all and and

for and any ...

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