 Summary
 I am working through J. H. Wiliamson’s Book on Lebesgue Integration on my own and have come across a proof I find rather “sketchy.”
If someone can straighten out my logic or concur with the presence of a mistake in the proof (even though the conclusion is correct, of course), I would be much obliged.
I’m looking at the proof of the corollary near the middle of the page (image of page attached below). I simply don’t find that the set, for instance, A_1 is finite, for if n=1, then wouldn’t it contain the infinite sequence of elements (writing only one memeber of each equivalence class of the rationals): 0/1, 1/1, 1/2, 1/3,...,2/3,...?
I understand the structure of the proof it uses the theorem presented above it, which proves that the union of countably infinite sets is countably infinite. I just don’t find how the particular portion of the statement of the proof mentioned above is accurate. Did the author, perhaps, mean to write “positive rationals, R_0?” But, if so, then why the inclusion of the absolute value in the equation governing the property of inclusion for the indexed sets?
I’m looking at the proof of the corollary near the middle of the page (image of page attached below). I simply don’t find that the set, for instance, A_1 is finite, for if n=1, then wouldn’t it contain the infinite sequence of elements (writing only one memeber of each equivalence class of the rationals): 0/1, 1/1, 1/2, 1/3,...,2/3,...?
I understand the structure of the proof it uses the theorem presented above it, which proves that the union of countably infinite sets is countably infinite. I just don’t find how the particular portion of the statement of the proof mentioned above is accurate. Did the author, perhaps, mean to write “positive rationals, R_0?” But, if so, then why the inclusion of the absolute value in the equation governing the property of inclusion for the indexed sets?
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