Shreyas’ Notes

Elements of Analysis

MATH 302

spring, freshman year

a statement is a sentence that is either true or false.

[see the comp182 logic section]

ABA \rightarrow B

Disjoined conclusions §

[see the comp182 logic section]

P(QR)P \rightarrow (Q \vee R)

Sets §

[see the comp182 sets section]

Indexed families of sets §

xn=1SnnxN:xSnxx \in \bigcup_{n=1}^{\infty} S_n \leftrightarrow \exists n_x \in \mathbb{N} : x \in S_{n_x}

xn=1SnnN:nSnxx \in \bigcap_{n=1}^{\infty} S_n \leftrightarrow \forall n \in \mathbb{N} : n \in S_{n_x}

Induction §

[see the comp182 induction section]

Real vs Rational numbers §

Upper and lower bounds §

A number uRu \in \mathbb{R} is an upper bound of SS if for any xSx \in S, uxu \geq x. We say that SS is bounded from above.

Supremum: least upper bound.

If uu is the supremum of SS and uSu \in S, uu is the maximum of SS.

A number lRl \in \mathbb{R} is a lower bound of SS if for any xSx \in S, lxl \leq x. We say that SS is bounded from below.

Infimum: greatest lower bound.

If ll is the infimum of SS and lSl \in S, ll is the minimum of SS.

Let SRS \subseteq \mathbb{R} and let uRu \in \mathbb{R} be an upper bound of SS. uu is the supremum of SS iff:

Least upper bound property (aka completeness property): Every non-empty subset of the real numbers that is bounded from above has a least upper bound that is a real number.

Bounded and nested intervals §

Archimedian property: For every real number xx, there exists a natural number nxn_x such that nx>xn_x > x.

Successor axiom: If nNn \in \mathbb{N}, n+1n + 1 exists and is a natural number.

A family of sets SnS_n indexed by the natural numbers is nested iff Sn+1SnnNS_{n + 1} \subseteq S_n \forall n \in \mathbb{N}.

Nested intervals property: If InI_n is a nested family of non-empty, closed, bounded[1] intervals in R\mathbb{R}, then n=1In\bigcap_{n = 1}^\infty I_n is non-empty.

Bounded Infinite sets have cluster points §

Neighbourhoods: Given xRx \in \mathbb{R} and ϵ>0\epsilon > 0, the ϵ\epsilon-neigbourhood is the open interval (xϵ,x+ϵ)(x - \epsilon, x + \epsilon).

A real number xx ix a cluster point [2] of a set SRS \subseteq \mathbb{R} if (xϵ,x+ϵ)(x - \epsilon, x + \epsilon) contains a non-xx element of SS for all ϵ>0\epsilon > 0.

Cluster points of a set need not be elements of the set.

Bolsano-Weierstrass Theorem: Every bounded, infinite set of real numbers has a cluster point.

Functions §

Given sets XX and YY, a function ff from XX to YY is denoted f:XYf : X \rightarrow Y, and it is a relation between XX and YY such that every xXx \in X has exactly one yYy \in Y such that f(x)=yf(x) = y.

XX is the domain and YY is the co-domain of ff.

f=gf = g iff they have the same domain and f(x)=g(x)f(x) = g(x) for all xx in their domain.

Let f:XYf: X \rightarrow Y, and AXA \subseteq X. The restriction of ff to AA is the function fA:AYf_A : A \rightarrow Y defined by fA(a)=f(a)aAf_A(a) = f(a) \forall a \in A.

Let f:XYf: X \rightarrow Y be a function. Given AXA \subseteq X, the image of AA under ff is the subset f(a)Yf(a) \subseteq Y given by f(a)={yY:aA:f(a)=y}f(a) = \{y \in Y : \exists a \in A : f(a) = y\}.

Open sets §

A set URU \subseteq \mathbb{R} is open if for every xUx \in U, there exists ϵx>0\epsilon_x > 0 such that (xϵx,x+ϵx)u(x - \epsilon_x, x + \epsilon_x) \subseteq u.

The intersection of any finite family of open sets is open.

Closed sets §

A set FRF \subseteq \mathbb{R} is closed if FF contains all of its cluster points.

If a set has no cluster points, it is closed.

Continuity §

ff is continuous if and only if for every x0x_0 in the domain: for every ϵ>0\epsilon > 0, there exists δ>0\delta > 0 such that for all xRx \in \mathbb{R}, if xx0<δ|x - x_0| < \delta, then f(x)f(x0)<ϵ|f(x) - f(x_0)| < \epsilon.

A function f:RRf : \mathbb{R} \rightarrow \mathbb{R} is continuous if for every open set uu in the codomain of ff, its pre-image f1(u)f^{-1}(u) is also open.

Sequences §

a sequence is a function NR\mathbb{N} \rightarrow \mathbb{R}.

a sequence converges to a real number LL if for every ϵ\epsilon-neighbourhood of LL, the sequence eventually remains in the ϵ\epsilon-neighbourhood.

(ϵ>0,NϵN:nNϵ,xnL<ϵ)xn(Lϵ,L+ϵ)(\forall \epsilon > 0, \exists N_\epsilon \in \mathbb{N} : \forall n \geq N_\epsilon, |x_n - L| < \epsilon) \Leftrightarrow x_n \in (L - \epsilon, L + \epsilon)

a sequence can have at most one limit.

Squeeze theorem: if limln=L\lim l_n = L, limun=L\lim u_n = L, and lnxnunl_n \leq x_n \leq u_n, then limxn=L\lim x_n = L.

A real number LL is a cluster point of a set SS of real numbers iff there exists a sequence of elements of SS, all different from LL, that converges to LL.

A set FRF \subseteq \mathbb{R} is closed iff for an arbitrary convergent sequence xnx_n in FF, the limit limxn\lim x_n is an element of FF.

A sequence is monotone if it is either weakly increasing, or weakly decreasing.

Every bounded, monotone sequence of real numbers converges.

Given a sequence (xn)(x_n), a subsequence is a sequence (yn)(y_n) defined by yk=xnky_k = x_{n_k} for some choice of natural numbers n1<n2<n_1 < n_2 < \cdots.

Bolzano-Weierstrass Theorem: Every bounded sequence of real numbers has a convergent subsequence.

A sequence is a cauchy sequence if the terms of the sequence are “eventually” close to each other.

ϵ>0,NN:m,nN,xmxn<ϵ\forall \epsilon > 0, \exists N \in \mathbb{N} : \forall m, n \geq N, |x_m - x_n| < \epsilon

A sequence of real numbers is convergent if and only if it is a Cauchy sequence.

A set SRS \subseteq \mathbb{R} is disconnected if there exist sets U,VSU, V \subseteq S such that:

  1. UV=SU \cup V = S
  2. UU \neq \varnothing and VV \neq \varnothing
  3. UV=U \cap V = \varnothing
  4. UU and VV are open in SS

Given DRD \subseteq \mathbb{R}, if f:DRf : D \rightarrow \mathbb{R} is continuous, then the image of any connected set is continuous.

Extreme value theorem: Suppose f:[a,b]Rf: [a, b] \rightarrow \mathbb{R} is continuous. Then f([a,b])={f(x):x[a,b]}f([a, b]) = \{f(x) : x \in [a, b]\} has a maximum and a minimum.

A set KRK \subseteq \mathbb{R} is compact if it is closed and bounded.

Given DRD \subseteq \mathbb{R}, if f:DRf : D \rightarrow \mathbb{R} is continuous, then the image of every compact set is compact.

Covering property (aka Heine-Borel theorem)" KK is closed and bounded if and only if every open cover of KK has a finite subcover.

  1. from above and from below ↩︎

  2. aka “accumulation point” or “limit point” ↩︎