The extended real line is homeomorphic (isomorphic) to the unit interval [0,1] so there are no surprises there, the maximum is infinity and it achieves this maximum.
This does not work for the real line, because plus and minus infinity do not exist there.
It is equivalent to asking about...
I think that you are slightly overthinking this, the real part of a complex number is just that - a real number.
And the modulus of a real number is just its absolute value, and a real number is always less than its absolute value.
I think you will find the proof in Rudin as a series of questions at the end of the chapter, if I recall correctly.
It begins with the notions of separability and of a base.
Ah thank you for the extremely prompt and enlightening replies, I am still learning and what seems obvious to most is rather opaque to me. My mistake here, was I think, taking MVT at its boring face value, rather than the practical information it gives us from the derivative be it neg, pos or 0...
Hi, I've been watching the MIT lectures on single variable calculus, and whilst proving FTC, he mentions that we since we know that: <$> f'(x) = g'(x) </$>, then by MVT we know that <$> f(x) = g(x) + C </$>.
I have tried searching for somewhere where this implication is spelled out for me...
Hello all, I have recently been wondering whether there is a way to determine a fraction for which the decimal expansion is a cycle of n numbers?
I would like to be able to work this out myself, but I can't wait until I start my mathematics degree. So any help would be greatly appreciated...