Convergence of monotone nets - Mathematics Stack Exchange In sequences of real numbers, we have a monotone convergence result: If an+1 ≥ an a n + 1 ≥ a n and bounded, then an a n converges to it's supremum The proof seems to work also in the net case My question is given that our net is not into the reals but a general linearly ordered space, and it is a monotonically increasing and bounded, can we say that such always converges in the order
Monotone convergence theorem for series (basic proof) My question is how to prove monotone convergence theorem for infinite series without more advanced technique like counting measure I see this used a lot But looking through books like Rudin, the
monotone class theorem, proof - Mathematics Stack Exchange Green Line: The monotone class generated by A A, which we call M M, is the smallest monotone class containing A A, meaning no other monotone class containing A A is properly contained inside M M
Strong convexity and strong monotonicity of the sub-differential However f f is not C1 C 1, so the reciprocal of Baillon Haddad is false Moreover this is a counter example to the original question because if the original question had a positive answer, g:= f∗ g:= f ∗ would have a strongly monotone subgradient hence be strictly convex and its conjugate g∗ = f∗∗ = f g ∗ = f ∗ ∗ = f would be C1