Let $S$ be a set. Recall a partition of $S$ is a collection of non-empty pairwise disjoint subsets with union $S$. Any partition $P$ of $S$ gives a relation $a\sim_P b \overset{\text{def}}{\iff} (a,b \text{ are in same element of }P).$ One naturally wonders what $\lbrace\sim_P \, : P\text{ a partition of }S\rbrace$ is. Some thought reveals it ... Read more 03 Aug 2021 - less than 1 minute read
Consider $ z^d + a_{d-1} z^{d-1} + \ldots + a_1 z + a_0 $ with each $ a_j \in \mathbb{C} $. Any root $ c \in \mathbb{C} $ must satisfy $ | c | \leq 1 + | a_{d-1} | + \ldots + | a_0 | $ (i.e. must be in the closed disk $ | z | \leq 1 + \sum | a_j | $) Proof is easy. Roots with $ | c | < 1 $ will obviously satisfy. For those with $ | c | \geq... Read more 03 Aug 2021 - less than 1 minute read
Here is one approach (another being this). Let $ (x_n) $ be a sequence from $ [a,b] $. Then it has an accumulation point $ c \in [a,b] $ : Let’s call an interval $ [p,q] $ “good” if $ x_n \in [p,q] $ for infinitely many $ n $. Break $ I_0 := [a,b] $ into two equal parts $ [a, \frac{a+b}{2}] $ and $[\frac{a+b}{2}, b] $. Atleast one of these mu... Read more 03 Aug 2021 - 1 minute read