Math UA 325: Analysis
New York University
Short study guide
May 2, 2023
Number systems: N,Z,Q.
Operations on sets: union, intersections, complement, de Morgan’s law,
Cartesian product.
Mathematical induction.
Functions: domain, codomain, graph, image, preimage, composition. In-
jectivity, surjectivity, bijectivity, inverse function. Interations between im-
age/preimage and intersection/union.
Equivalence relations, cardinality, countability. N,Z,Q are countable, while
P(N) is uncountable. Finite sets.
Ordered sets, least upper bound (supremum) and greatest lower bound (in-
fimum). Q does not satisfy the least upper bound property.
Real numbers R: ordered field with the least upper bound property that
contains Q. Irrational numbers, e.g.
√
2. Archimedean property and den-
sity of Q in R.
Basic properties of sup and inf. Maximum and minimum. Absolute value,
triangle inequality (including the version with n terms), reverse triangle in-
equality. Bounded functions.
Sequences: bounded sequences, convergent and divergent sequences. Unique-
ness of the limit (when it exists). Covergent sequences are bounded. Mono-
tone sequences. Tails and subsequences. Squeeze lemma.
Basic properties of limits. Ratio test. Bolzano-Weierstrass Theorem: a
bounded sequence has a convergent subsequence. Cauchy sequences. A se-
quence is Cauchy if and only if it is convergent.
Series: partial sums. Geometric series
∑
n r
n. “Sanity check”: if
∑
n xn con-
verges then limn xn = 0. Divergence of the harmonic series
∑
n
1
n . Absolute
1
convergence and conditional convergence for series with xn not necessarily
nonnegative. Inequality |∑n xn| ⩽ ∑n |xn| for an absolutely convergent
series.
Convergence tests: Comparison test. p-series test. Ratio test. Root test.
Alternating series test.
Limits of functions limx→c f(x). Basic properties (sum, product, quotient
when there is no division by zero). The fact that limx→c f(x) = L if and
only if for every sequence xn → c we have limn f(xn) = L. Left and right
limits.
Continuity: Definition of continuity. Continuity of polynomials. Basic prop-
erties (sums, products of continuous are continuous; quotients as well wher-
ever the denominator does not vanish). Continuous functions on a closed
bounded interval achieve a global maximum and minimum. Intermediate
Value Theorem. Odd degree polynomials have a root. Uniform continuity.
A continuous function on a closed bounded interval is uniformly continuous.
Derivatives: Definition of derivative. Slope of tangent line to the graph.
Differentiable functions are continuous. Basic properties (sums, products of
differentiable are differentiable; quotients as well wherever the denominator
does not vanish). Formulas for the derivative of a sum, of a rescaling, of
a product (Leibniz rule), of a quotient, Chain Rule, derivative of the inverse.
Relative maxima and minimal of a differentiable function are critical points
(i.e. f ′(x) = 0). Rolle’s Theorem and the Mean Value Theorem. A differ-
entiable function on an interval with zero derivative everywhere is constant.
Monotonicity in terms of sign of the derivative. Taylor’s Theorem.
Integrals: Partitions of [a, b] and upper and lower Darboux sums of a bounded
function. Upper and lower integrals and their behavior under refinement of
P . Definition of integrability. Basic inequality m(b− a) ⩽ ∫ ba f ⩽M(b− a)
where m ⩽ f(x) ⩽ M for all x ∈ [a, b]. Additivity of domain (chop). Lin-
earity of the integral and monotonicity. Fundamental Theorem of Calculus
(versions 1 and 2), change of variable formula, integration by parts.
Logarithm, Exponential and their basic properties.
Sequences of functions: pointwise convergence, uniform convergence. Uni-
2
form limit of continuous functions is continuous. Uniform limit of integrable
functions is integrable, and in this case taking limit commutes with integra-
tion.