@Jakobian: Thanks! Another question: if $a_n>b_n$ for each $n \in \mathbb{N}$, can we say that $\sup_{n\in\mathbb{N}}a_n > \sup_{n\in\mathbb{N}} b_n$? I think it is true because from $\sup_{n\in\mathbb{N}} a_n \ge a_n>b_n$ we deduce that $\sup_{n\in\mathbb{N}}a_n$ is an upper bound for $b_n$, hence $\sup_{n\in\mathbb{N}}a_n \ge \sup_{n\in\mathbb{N}} b_n$ because by definition the supremum is the least upper bound.
If by the sake of contradiction we assume $\sup_{n\in\mathbb{N}}a_n = \sup_{n\in\mathbb{N}}b_n$, we have $\sup_{n\in\mathbb{N}}b_n=\sup_{n\in\mathbb{N}}a_n \ge a_n>b_n$, so $\sup…
If by the sake of contradiction we assume $\sup_{n\in\mathbb{N}}a_n = \sup_{n\in\mathbb{N}}b_n$, we have $\sup_{n\in\mathbb{N}}b_n=\sup_{n\in\mathbb{N}}a_n \ge a_n>b_n$, so $\sup…
Our professor recently taught us minimal polynomial in a Linear Algebra course. At the end of his lecture, he wrote on board just how does a Jordan matrix looks like and told us to consider that as the apparent "definition" for the same. He didn't teach us anything in 'detail' about Jordan matric...
I see that we've recently just created the tag geometric-topology. Considering that the subject has its own arXiv subject code, I don't object to the tag's existence. But for me, geometric topology sort of lies in the fuzzy area between differential topology, differential geometry, and low dimens...