- #1

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I need help with some further aspects of the proof of Lemma 3.3 ... ... in order to fully understand the proof ...

The statement and proof of Lemma 3.3 of L&S reads as follows:

__QUESTION 1__In the above proof by L&S we read the following:

" ... ... Let n be fixed. There are integers \(\displaystyle p_j, q_j \) such that

\(\displaystyle \frac{p_j - 1}{n} \lt a_j \leq \frac{p_j}{n} \text{ and } \frac{q_j - 1}{n} \lt b_j \leq \frac{q_j}{n} \)

... ... ... ... "

I tried several numerical examples ... and the examples indicated the above was true ...

... BUT ... ...

How do we prove that there exist integers \(\displaystyle p_j, q_j \) such that

\(\displaystyle \frac{p_j - 1}{n} \lt a_j \leq \frac{p_j}{n} \text{ and } \frac{q_j - 1}{n} \lt b_j \leq \frac{q_j}{n} \)

... ... ... ...

__QUESTION 2__In the above proof by L&S we read the following:

" ... ... One can easily see that a cube \(\displaystyle [ \frac{ i_1 - 1 }{n} , \frac{ i_1 }{n} ] \ \times \ldots \ \ldots \times \ [ \frac{ i_p - 1 }{n} , \frac{ i_p }{n} ] \) intersects the closure of R ( i.e. R itself) if \(\displaystyle p_j \leq i_j \leq q_j \) (j = 1, ... ... p) ... ...

I am unsure of how to visualize this ...

... ... ... can someone please explain how to formally and rigorously prove that a cube \(\displaystyle [ \frac{ i_1 - 1 }{n} , \frac{ i_1 }{n} ] \ \times \ldots \ \ldots \times \ [ \frac{ i_p - 1 }{n} , \frac{ i_p }{n} ] \) intersects the closure of R ( i.e. R itself) if \(\displaystyle p_j \leq i_j \leq q_j \) (j = 1, ... ... p) ... ...

Help with the above two questions will be much appreciated ...

Peter

__NOTE:__To make sense of Lemma 3.3 readers of the above post will need access to pages 95-97 of L&S so I providing this text as follows:

Hope that helps,

Peter