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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