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I am focused on Chapter 16: Lebesgue Measure ... ...

I need help with an aspect of the proof of Proposition 16.1 ...

Proposition 16.1 and its proof read as follows:

In the above text from Carothers we read the following:

" ... ... But now, by expanding each \(\displaystyle J_k\) slightly and shrinking each \(\displaystyle I_n\) slightly, we may suppose that the \(\displaystyle J_k\) are open and the \(\displaystyle I_n\) are closed. ... "Can someone please explain how Carothers is expecting the \(\displaystyle J_k\) to be expanded and the \(\displaystyle I_n\) to be shrunk ... and further, why the proof is still valid after the \(\displaystyle J_k\) and \(\displaystyle I_n\) have been altered in this way ... ...

Help will be appreciated ...

Peter