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