My understanding is that in the original theorem, we want to prove
∃y.∀x.a≤x∧x≤b⇒|fx|≤yIf we contradict this statement, it becomes:
∀y.∃x.(a≤x∧x≤b)∧y<|fx|which can be further simplified to the following form:
∀k.(a≤(g m)∧(g m)≤b)∧m<|(f(g m))
where m is a positive integer and (g_m)...
Thanks for the clarification and I understand the point. But still I am not very convinced with the proof of Theorem 4 in the cited document above.
The proof utilizes the above statement to conclude that two distinct values of the sequence ($a_k$), say $a_K$ and $a_K′$ would exist in a...
I have across the following argument, which seems wrong to me, in a larger proof (Theorem 4 on page 9 of the document available at http://www.whitman.edu/mathematics/SeniorProjectArchive/2011/SeniorProject_JonathanWells.pdf). I would appreciate if someone can shed light on why this is true...