Real Analysis Review

Everything from the section, arranged for lookup rather than for reading straight through: the definitions, the named theorems, the counterexamples that force the definitions to be what they are, and the notation.

The one axiom separating $\mathbb{R}$ from $\mathbb{Q}$:

Every nonempty set of reals that is bounded above has a least upper bound.

$\mathbb{Q}$ fails it: $\{x \in \mathbb{Q} : x^2 < 2\}$ is bounded above with no rational supremum. Every existence theorem in the section traces back to this axiom.

The $\varepsilon$-characterization, which is how supremum is used in practice: $s = \sup S$ iff $s$ is an upper bound and for every $\varepsilon>0$ there is $x \in S$ with $x > s - \varepsilon$.

Consequences: the Archimedean property, density of $\mathbb{Q}$ and of the irrationals, the nested interval property, and the existence of $\sqrt{2}$.

Name	Statement
Monotone convergence	A bounded monotone sequence converges
Bolzano–Weierstrass	Every bounded sequence has a convergent subsequence
Heine–Borel	A subset of $\mathbb{R}$ is compact iff it is closed and bounded

These are three faces of completeness, and each is used to build a point out of nothing but bounds.

$a_n \to L$ means: for every $\varepsilon>0$ there is $N$ with $|a_n - L|<\varepsilon$ for all $n\ge N$. Written in that order, $N$ depends on $\varepsilon$ only.

• Limits are unique.
• Convergent sequences are bounded; bounded sequences need not converge.
• Limit laws hold for sums, products, quotients with nonzero limit.
• Squeeze theorem: $a_n \le b_n \le c_n$ with $a_n, c_n \to L$ forces $b_n \to L$.

A monotone bounded sequence converges, to the sup or inf of its range. This needs no candidate limit, which is why it is the first tool to reach for.

A sequence is Cauchy if its terms eventually cluster together: for every $\varepsilon$ there is $N$ with $|a_n - a_m|<\varepsilon$ for all $n,m\ge N$.

$$\text{Cauchy} \iff \text{convergent} \quad \text{(in } \mathbb{R}\text{)}$$

The forward direction is easy; the converse is completeness in yet another form. In $\mathbb{Q}$ it fails, which is exactly what ”$\mathbb{Q}$ has holes” means.

Not sufficient: consecutive terms getting close. $a_n = \sqrt{n}$ has $a_{n+1}-a_n \to 0$ and diverges. The Cauchy condition must hold for all pairs beyond $N$, not just neighbours.

• If $a_n \to L$ then every subsequence converges to $L$.
• Two subsequences with different limits proves divergence - the cleanest divergence proof there is.
• $\limsup$ and $\liminf$ are the largest and smallest subsequential limits, and always exist.

$\sum a_n$ converges when the partial sums do. That is the definition; everything else is a test.

• nth-term test. $a_n \to 0$ is necessary, never sufficient.
• Geometric. $\sum r^n$ converges iff $|r|<1$.
• $p$-series. $\sum 1/n^p$ converges iff $p>1$. The harmonic series $p=1$ diverges.
• Comparison, ratio, root, integral tests for positive series.
• Alternating series test for decreasing terms with $a_n\to0$.

Absolute versus conditional. Absolute convergence implies convergence. A conditionally convergent series such as $\sum(-1)^{n+1}/n$ can be rearranged to sum to anything, which is Riemann’s rearrangement theorem, while an absolutely convergent series is rearrangement-proof.

• Open: every point has a neighbourhood inside the set. Closed: the complement is open, equivalently the set contains all its limit points.
• $\varnothing$ and $\mathbb{R}$ are both open and closed. $[0,1)$ is neither. “Not open” does not mean closed.
• Arbitrary unions of open sets are open; only finite intersections are. Dually for closed sets, and $\bigcap(-1/n,1/n)=\{0\}$ shows why the finiteness matters.
• Compact means every open cover has a finite subcover, and by Heine–Borel that is the same as closed and bounded.
• Compactness is what makes $[a,b]$ special: $(0,1]$ and $[0,\infty)$ each fail one condition, and each breaks a theorem.

$$\lim_{x\to c}f(x)=L: \quad \forall\varepsilon>0\ \exists\delta>0:\ 0<|x-c|<\delta \implies |f(x)-L|<\varepsilon$$

The value $f(c)$ plays no part, and $c$ need not be in the domain.

$f$ is continuous at $c$ iff $c$ is in the domain and $\lim_{x\to c}f(x)=f(c)$. Equivalently, every sequence $x_n\to c$ has $f(x_n)\to f(c)$ - the sequential criterion, which is the fastest way to prove discontinuity.

For $f$ continuous on $[a,b]$:

Theorem	Conclusion
Boundedness	$f$ is bounded
Extreme value	$f$ attains a maximum and a minimum
Intermediate value	$f$ takes every value between $f(a)$ and $f(b)$
Uniform continuity	$f$ is uniformly continuous

All four need both closed and bounded. $f(x)=1/x$ on $(0,1]$ is unbounded; $f(x)=x$ on $[0,\infty)$ attains no maximum; $\sin(1/x)$ on $(0,1]$ is continuous and not uniformly continuous.

Uniform continuity moves the quantifier: one $\delta$ for the whole domain rather than one per point. Lipschitz implies uniformly continuous implies continuous, and both implications are strict - $\sqrt{x}$ on $[0,1]$ is uniformly continuous and not Lipschitz.

$$f'(c)=\lim_{h\to0}\frac{f(c+h)-f(c)}{h}$$

Differentiable implies continuous; the converse fails at every point for $|x|$ at 0, and Weierstrass built a function continuous everywhere and differentiable nowhere.

Interior extremum theorem: an interior extremum of a differentiable function has $f'=0$. That plus EVT gives Rolle, and Rolle after subtracting the secant gives the mean value theorem.

MVT consequences, all used freely in calculus and proved only here:

• $f'=0$ on an interval $\implies f$ constant. This is what makes “+C” a theorem.
• $f'>0 \implies f$ strictly increasing.
• $|f'|\le K \implies f$ is Lipschitz with constant $K$.

Taylor’s theorem is Rolle applied $n+1$ times:

$$f(x)=P_n(x)+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}$$

an equality with only $\xi$ unspecified. It gives the usable bound $|R_n| \le \frac{K}{(n+1)!}|x-a|^{n+1}$.

Smooth is not analytic. $e^{-1/x^2}$ has every derivative zero at 0, so its Taylor series converges everywhere to the wrong function. The series converging and the series converging to $f$ are different claims, and the second is exactly $R_n \to 0$.

For bounded $f$, partitions give $L(f,P)=\sum m_i\Delta x_i$ and $U(f,P)=\sum M_i\Delta x_i$. Refining raises $L$ and lowers $U$, and every lower sum is below every upper sum.

Integrable means $\sup_P L = \inf_P U$. Equivalently (Riemann criterion), for each $\varepsilon$ some $P$ has $U(f,P)-L(f,P)<\varepsilon$.

• Continuous $\implies$ integrable, via uniform continuity on the compact interval.
• Monotone $\implies$ integrable, even with countably many jumps.
• The Dirichlet function (1 on rationals, 0 on irrationals) is bounded and not integrable: $L=0$ and $U=1$ for every partition.

Fundamental theorem, part one. $F(x)=\int_a^x f$ is continuous whenever $f$ is integrable, and $F'(c)=f(c)$ wherever $f$ is continuous. Integration smooths.

Fundamental theorem, part two. $\int_a^b f = F(b)-F(a)$ for any antiderivative $F$, proved by applying the MVT on each subinterval and telescoping.

Pointwise: $N$ may depend on $x$. Uniform: one $N$ for all $x$, equivalently $\|f_n-f\|\to0$.

Preserved by uniform convergence	Not preserved
Continuity of the limit	Differentiability
The value of the integral	Anything about $f_n'$
The uniform Cauchy property	-

• $x^n$ on $[0,1]$: continuous terms, discontinuous limit, $\|f_n-f\|=1$ always. Uniform on $[0,a]$ for $a<1$.
• $\frac{\sin(nx)}{\sqrt n} \rightrightarrows 0$ while $f_n'=\sqrt n\cos(nx)$ diverges.
• A spike of height $n$ and width $1/n$: pointwise limit 0, every integral 1.
• Weierstrass M-test. $|u_n(x)|\le M_n$ with $\sum M_n<\infty$ gives uniform convergence, and licenses term-by-term integration of power series.

Theorem	Statement
Completeness axiom	Every nonempty set bounded above has a supremum
Archimedean property	For any real $x$ there is a natural $n > x$
Monotone convergence	A bounded monotone sequence converges
Bolzano–Weierstrass	Every bounded sequence has a convergent subsequence
Cauchy criterion	A sequence of reals converges iff it is Cauchy
Riemann rearrangement	A conditionally convergent series can be rearranged to any sum
Heine–Borel	Compact in $\mathbb{R}$ = closed and bounded
Extreme value	A continuous function on a compact interval attains its extremes
Intermediate value	A continuous function on $[a,b]$ takes every intermediate value
Uniform continuity theorem	Continuous on a compact set implies uniformly continuous
Rolle	Equal endpoint values give an interior critical point
Mean value theorem	$f'(c)$ equals the average rate for some interior $c$
Taylor	$f = P_n + f^{(n+1)}(\xi)(x-a)^{n+1}/(n+1)!$
Riemann criterion	Integrable iff some partition makes $U - L$ arbitrarily small
Fundamental theorem of calculus	$F' = f$ where $f$ is continuous, and $\int_a^b f = F(b)-F(a)$
Weierstrass M-test	Uniform bounds with a convergent sum give uniform convergence

They are worth memorizing as a set, because each one is the reason some hypothesis cannot be dropped.

Object	Shows
$\{x\in\mathbb{Q}:x^2<2\}$	$\mathbb{Q}$ is not complete
$(-1)^n$	Bounded need not imply convergent
$\sqrt{n}$	Consecutive terms close does not imply Cauchy
$\sum 1/n$	$a_n\to0$ does not imply convergence
$\sum(-1)^{n+1}/n$	Convergent but not absolutely; rearrangeable
$[0,1)$	A set that is neither open nor closed
$\bigcap(-1/n,1/n)$	Infinite intersections of open sets need not be open
$1/x$ on $(0,1]$	Non-compact domain breaks boundedness
$\sin(1/x)$ on $(0,1]$	Continuous, not uniformly continuous
$x^2$ on $\mathbb{R}$	Continuous, not uniformly continuous on an unbounded set
$\sqrt{x}$ on $[0,1]$	Uniformly continuous, not Lipschitz
$\lvert x\rvert$ at 0	Continuous, not differentiable
$e^{-1/x^2}$	Smooth, not analytic
Dirichlet function	Bounded, not Riemann integrable
$x^n$ on $[0,1]$	Pointwise limit of continuous functions can jump
$\sin(nx)/\sqrt n$	Uniform convergence says nothing about derivatives
Height-$n$ spike of width $1/n$	Limit and integral do not always commute

• Confusing sup with max. $\sup[0,1) = 1$ and there is no maximum.
• Letting $N$ depend on $x$ in a uniform statement, or on $n$ in a sequence statement. Quantifier order is the content.
• Thinking $a_{n+1}-a_n\to0$ makes a sequence Cauchy. $\sqrt n$ refutes it.
• Using the nth-term test to prove convergence. It can only prove divergence.
• Rearranging a conditionally convergent series.
• Assuming “not open” means closed.
• Applying EVT or IVT on an open or unbounded interval.
• Forgetting that $f(c)$ is irrelevant to $\lim_{x\to c}f(x)$.
• Treating differentiable and continuous as interchangeable.
• Reading Taylor’s theorem as an approximation. It is an equality; the remainder is exact.
• Assuming a smooth function equals its Taylor series.
• Concluding integrability from boundedness. The Dirichlet function is bounded and not integrable.
• Interchanging a limit with an integral or a derivative without uniformity.
• Expecting uniform convergence of $f_n$ to say something about $f_n'$.