A real function is continous on $[a,b]$ such that either $f(a) > 0 >f(b)$ or $f(a) < 0 < f(b)$, then prove that $f(c) = 0$ for some $c \in (a,b)$. Proof :- We divide $[a,b]^*$ in $H$ equal parts, where $H$ is a positive infinite hyperinteger. We get, $$a, a +\delta, a+2\delta,\ ... \ , ...

A search result for Mean Value Theorem gives us 2715 results, and results on the page are like ones I think we can include in the tag. The theorem is an important result in calculus, and questions relating to its applications, proofs. I think it would be useful if could have the tag, as it can gr...

Fix an Abelian category $\mathscr{A}$, and a homological spectral sequence $(E_{p,q}^r)$, so that $E^{r+1}\cong H(E^r)$, then, according to the ncatlab, we say that $E^r$ converges to $E^\infty$ if for all $(p,q)$, there exists some $r_0$ such that $r\ge r_0$ implies that $E^{r}_{p,q}\cong E^\inf...

Let $\mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $\bigcap\mathscr{F}\ne\emptyset$. Prove that $\bigcup\mathscr{F}$ is connected. If $\bigcup\mathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty subsets $A,B$. Let $x$ be a...