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

I am reading N. L. Carothers' book: "Real Analysis". ... ...

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 ##J_k## slightly and shrinking each ##I_n## slightly, we may suppose that the ##J_k## are open and the ##I_n## are closed. ... "

Can someone please explain how Carothers is expecting the ##J_k## to be expanded and the ##I_n## to be shrunk ... and further, why the proof is still valid after the ##J_k## and ##I_n## have been altered in this way ... ...

Help will be appreciated ...

Peter

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 ##J_k## slightly and shrinking each ##I_n## slightly, we may suppose that the ##J_k## are open and the ##I_n## are closed. ... "

Can someone please explain how Carothers is expecting the ##J_k## to be expanded and the ##I_n## to be shrunk ... and further, why the proof is still valid after the ##J_k## and ##I_n## have been altered in this way ... ...

Help will be appreciated ...

Peter