From 7fc092422474f83f54c3dd8684db9074e024ca90 Mon Sep 17 00:00:00 2001
From: 8oldr <8oldr@proton.me>
Date: Tue, 3 Nov 2026 14:04:37 +0200
Subject: First hour

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+#set document(
+  title: [Lecture 3],
+  author: "Gidon Rosalki",
+  date: auto,
+)
+#set heading(numbering: "1.")
+#set text(font: "New Computer Modern")
+#set page(margin: (
+  bottom: 0.5cm,
+  right: 1.5cm,
+  left: 1.5cm,
+  top: 1.5cm,
+))
+#set list(indent: 10pt)
+#set enum(indent: 10pt)
+
+#show title: set align(center)
+#show link: set text(rgb("005eff"))
+#show link: underline
+#show math.equation.where(block: true): set block(breakable: true)
+#title()
+#align(center)[
+  Gidon Rosalki \
+  2026-11-03
+]
+
+#let colourmaths(x, color) = text(fill: color)[$#x$]
+
+// author: eric1102 - https://sitandr.github.io/typst-examples-book/book/typstonomicon/math_display.html
+// may slightly interfere with maths blocks
+#show math.equation: it => {
+  if it.body.fields().at("size", default: none) != "display" {
+    return math.display(it)
+  }
+  it
+}
+#let qquad = h(0.5em)
+
+= Introduction
+
+We will be speaking about Infi today! Hooray! Not Linear Algebra, but Infi! Praise be, and gird thy loins. It appears
+that this will be a recap of stuff we discussed in the bachelors degree courses, but I'm writing this paragraph at the
+start of the lecture, so that may yet change...
+
+= Limits, Sets, Continuity
+
+*Definition* $RR^d$: This is the $d$-dimensional euclidean vector space over $RR$.
+$
+  RR^d = {vec(x_1, x_2, dots.v, x_d) : x_1, dots, x_d in RR}
+$
+We define the standard basis, which has elements
+$
+  e_1 = vec(1, 0, dots.v, 0), quad e_2 = vec(0, 1, dots.v, 0), dots, quad e_d = vec(0, 0, dots.v, 1)
+$
+
+*Definition* Euclidean norm:
+$
+  norm(x) = sqrt(sum_(i=1)^(d)x_i^2)
+$
+Properties of $norm(dot.c)$:
+- Positive definite: $forall x qquad norm(x) >= 0, norm(x) = 0 <=> x = 0$
+- Triangle inequality: $forall x, y qquad norm(x + y) <= norm(x) + norm(y)$
+- Homogeneity: $forall alpha in RR, x in RR^d, qquad norm(alpha x) = abs(alpha) norm(x)$
+
+*Definition* (distance): The _distance_ between $x, y in RR^d$ is $d(x, y) = norm(x - y)$ \
+Properties of $d(dot.c, dot.c)$:
+- Positive definite: $d(x, y) >= 0, d(x, y) = 0 <=> x = y$
+- Triangle inequality: $forall x, y, z, qquad d(x, z) <= d(x, y) + d(y, z)$
+- Symmetry :$forall x, y qquad d(x, y) = d(y, x)$
+
+*Definition* (Open ball): The _open ball_ of radius $r > 0$, centred at $x in RR^d$ is the set
+$
+  B_r(x) = {y in RR^d : d(x, y) < r}
+$
+*Definition* (Closed ball): The _closed ball_ of radius $r > 0$, centred at $x in RR^d$ is the set
+$
+  overline(B_r)(x) = {y in RR^d : d(x, y) <= r}
+$
+
+== Limits
+
+*Definition* A sequence ${x_n}_(n = 1)^(infinity) subset.eq RR^d$ _converges_ to $x in RR^d$ if $forall i in [d],
+qquad lim_(n -> infinity) x_i^n = x_i$
+
+#pagebreak()
+*Proposition*: Let ${x^n}_(n = 1)^infinity subset.eq RR^d$. It converges to $x in RR^d$ *if and only if* $lim_(n ->
+infinity) norm(x^n - x) = 0$. We will note that this is arguing an alternate definition to the one above. \
+_Proof_: Let us assume $lim_(n_> infinity) x^n = x$:
+$
+  forall n, qquad norm(x^n - x) & = sqrt(sum_(i = 1)^(d) (x_i^n - x_i)^2) \
+                                & underbrace(-->, n -> infinity) 0
+$
+Which follows since all of squared, sum, and square root are continuous functions, and by definition $x_i^n - x_i$ tends
+to 0. \
+Let us assume that $norm(x^n -> x)$:
+$
+  sqrt((x_i^n - x_i)^2) & = abs(x_i^n - x_i) \
+                        & <= sqrt(sum_(i =1)^(d) (x_j^n - x_j)^2) \
+                      7 & underbrace(-->, n -> infinity) 0
+$
+
+*Definition* (Subsequence): Let ${x^n}_(n = 1)^(infinity) subset.eq RR^d$. A _subsequence_ is any sequence ${x^(n_k)}_(k =
+1)^(infinity)$, where ${n_k}_(k = 1)^infinity$ is a strictly increasing sequence in $NN$. \
+In short, a subsequence is an infinitely large subset of the original sequence, selected in increasing order of indices.
+If we consider $NN$ to be our sequence, then sub-sequences could be the even numbers, the odd numbers, the prime
+numbers, randomly selected numbers in increasing order, and so on.
+
+*Proposition*: Fix sequences ${x^n}_(n = 1)^(infinity), {y^n}_(n = 1)^(infinity) subset.eq RR^d$ and
+${alpha^n}_(n = 1)^(infinity) subset.eq RR$, with respective limits $x, y in RR^d, qquad alpha in RR$. Then:
++ $lim_(n -> infinity) (x^n + y^n) = x + y$
++ $lim_(n -> infinity) alpha_n x^n = alpha x$
++ Every subsequence of ${x^n}$ converges to $x$
+_Proof_:
++ For every $n$:
+$
+  norm(x^n + y^n - (x + y)) & overbrace(<=, triangle) norm(x^n - x) + norm(y^n - y) \
+                            & --> 0 + 0
+$
+
+= Sets
+
+*Definition* (closed sets): A set $A subset.eq RR^d$ is _closed_ if for every convergent sequence ${x^n}_(n = 1)^infinity subset.eq A$, their limit is also in $A$ (i.e. $lim_(n -> infinity) x^n in A)$. \
+*Example* (half spaces): Let $w in RR^d, qquad a in RR$, and define $H = {x in RR^d : lr(chevron.l x, w chevron.r) >=
+  a}$ where \ $lr(chevron.l x, w chevron.r) = sum_(i = 1)^(d) x_i w_i$. Suppose ${x^n}_(n = 1)^infinity subset.eq H$
+converges to $x$. We must prove that $x in F$, i.e. $lr(chevron.l x, w chevron.r) >= a$:
+$
+  lr(chevron.l x, w chevron.r) & = sum_(i = 1)^(d)x_i w_i \
+                               & = sum_(i = 1)^(d) w_i lim_(n -> infinity)x_i^n \
+                               & = lim_(n -> infinity) underbrace(
+                                   sum_(i = 1)^(d)w_i x_i^n, lr(
+                                     chevron.l x^n, w
+                                     chevron.r
+                                   ) >= a
+                                 ) \
+                               & >= a
+$
+#pagebreak()
+*Example* (sphere): $S^(d - 1) = {x in RR^d : norm(x) = 1}$. Suppose ${x^n}_(n = 1)^(infinity) subset.eq S^(d - 1)$
+that converges to $x$.
+$
+  norm(x) & = sqrt(sum_(i in [d])^() x_i^2) \
+          & = sqrt(sum_(i in [d])^()(lim_(n -> infinity)x_i^n)^2) \
+          & = lim_(n -> infinity) sqrt(sum_(i in [d])^() (x_i^n)^2) \
+          & = lim_(n -> infinity) norm(x^n) \
+          & = 1
+$
+Thus the sphere is a closed set.
+
+
+*Definition* (open set): A set $U$ is _open_ if $forall x in U qquad exists epsilon > 0 : B_epsilon (x) subset.eq U$.
+
+*Example* (open half spaces): Let $w in RR^d, a in RR$, and define $H = {x in RR^d : lr(chevron.l w, x chevron.r) > a}$.
+We claim that $H$ is open: Let $x in H$. Fix $epsilon > 0$, and suppose that $y in B_epsilon (x)$:
+$
+  lr(chevron.l y, w chevron.r) & = lr(chevron.l y - x + x, w chevron.r) \
+  & = lr(chevron.l y - x, w chevron.r) + underbrace(lr(chevron.l x, w chevron.r), > a) \
+  abs(lr(chevron.l y - x, w chevron.r)) & underbrace(<=, "Cauchy-Schwarz") norm(y - x) norm(w) \
+  & < epsilon norm(w) \
+  ==> lr(chevron.l y - x, w chevron.r) + lr(chevron.l x, w chevron.r) & > lr(chevron.l x, w chevron.r) - epsilon norm(w) \
+  lr(chevron.l x, w chevron.r) - epsilon norm(w) & > a \
+  <==> epsilon norm(w) & < lr(chevron.l x, w chevron.r) - a \
+  <==> epsilon & < (lr(chevron.l x, w chevron.r) - a) / norm(w)
+$
+
-- 
cgit