Here is one approach (another being this). Let $ (x_n) $ be a bounded sequence of reals. Thanks to Bolzano-Weierstrass, $ A := \lbrace \text{ accumulation points of }(x_n) \rbrace $ is non-empty. $ A $ is also bounded above, so $ a := \sup(A) $ exists, which we call $ \color{goldenrod}{\limsup x_n} $. $ a \in A $, i.e. $ a $ itself is an acc... Read more 04 Aug 2021 - 1 minute read
Let $ (x_n) $ be a sequence of reals. $ p \in \mathbb{R} $ is an $\color{goldenrod}{\text{accumulation point}}$ of the sequence if for every $ \epsilon > 0 $, $ x_n $ lies in $ (p-\epsilon, p+\epsilon) $ for infinitely many $ n $ (stricten “infinitely many” to “all but finitely many”, and we get definition of limit of a sequence). A subsequ... Read more 03 Aug 2021 - 1 minute read
Test, $ x^2 \geq 0 $. def square(x): return x**2 Read more 03 Aug 2021 - less than 1 minute read
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