$\forall$ has $\exists$ as a "left adjoint" in Topos Theory.
Let $M \xrightarrow{f} M' \xrightarrow{g} M''$ be sequence of $A$-modules and $A$-module homomorphisms. Now suppose we don't have $gf = 0$ or in other words no ability to compute traditional Homology at $M'$, but instead we have sort...
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Given a sequence $x_{n}$ and initial data that $0< a\leq x_{n}\leq b< \infty $,
for $a,b\in \mathbb{R}$.
I need to show that: $$\limsup \frac{1}{x_{n}}\cdot \limsup x_{n}\geq 1.$$
I think the the simplest way to do that is to show that $$\liminf \frac{1}{x_{n}}=\frac{1}{\limsup x_{n}}.$$
I s...
