Analysis I (Math 317) -- Fall 2006 -- Main Definitions and Theorems
Main Definitions
Limit of a sequence: we say the sequence an has limit L if for all eps > 0,
there exists a natural number N such that for all n >= N, |an - L| < eps.
Supremum of a set: we say B is the supremum of a set S of real numbers if (1) B >= x for all x in S
and (2) for all upper bounds M of S, B <= M.
Infimum of a set: we say A is the infimum of a set S of real numbers if (1) A <= x for all x in S
and (2) for all lower bounds m of S, A >= m.
limsup xn = limn to infinity sup { xn, xn+1, xn+2,...}
(and similarly for liminf)
Cardinality (countable, uncountable): a set S is countable if it can be put into
one-to-one correspondence with some subset of N; a set is uncountable if it is not countable.
Main Theorems
The triangle inequality (and junior)
The limit of a sequence is unique
A convergent sequence is bounded
Squeeze Theorem
If an <= bn for all n and an converges to a,
and bn converges to b, then a <= b (from HW)
Limit Laws (limit of sum, difference, product, quotient)
Epsilon-version of supremum or infimum
Every subset of R that is bounded above has a supremum
Monotone Convergence Theorem: every increasing sequence of real numbers that is bounded above has a limit
N,Z,Q are countable
R is uncountable
A union of a countable number of countable sets is countable
A Cartesian product of two countable sets is countable
Material for Test #2 is below here
Main Definitions
Subsequence: given a sequence xn, a subsequence is any xnk, where
n1 < n2 < ...
Cauchy sequence
Partial Sums of an infinite series
Convergence of a series of real numbers
Absolute convergence of a series of real numbers
Conditional convergence of a series of real numbers
Norm on a vector space
Br(x) (ball of radius r around x)
Limit point of a set
Cluster point of a set
Closed set
Open set
Interior of a set
Closure of a set
Boundary of a set
Open cover, finite subcover
Sequentially compact set
Compact set
Connected set
Cantor set
Dense
Main Theorems
Bolzano-Weierstrass Theorem
A sequence has a limit if and only if it is Cauchy
Divergence Test
The convergence of, and sum of, a geometric series
Integral Test
Comparison Test
Limit Comparison Test
Ratio Test
Root Test
Alternating Series Test
Absolute Convergence Theorem
A sequence in Rn converges if and only if the sequence in each component converges
Limit Laws still work in Rn as long as they make sense (i.e., only multiplication
and division rules if range is in R)
Any epsilon-ball is open
Finite intersection or arbitrary union of open sets is open
Finite union or arbitrary intersection of closed sets is closed
A set is closed if and only if its complement is open
Closure of A is smallest closest set containing A
Interior of A is largest open set contained in A
Closure of a closed set equals itself
Interior of an open set equals itself
Three ways of thinking about boundary of A: closure of A intersected with closure of A',
or closure of A minus interior of A, or x is in boundary of A if and only if every Br(x)
contains a point in A and a point not in A.
Borel-Lebesgue Theorem: A set is compact if and only if it is sequentially compact.
Heine-Borel Theorem: A subset of Rn is compact if and only if it is closed and bounded.
Test #3 (Chapters 5 (5.1--5.4) and 6 (6.1--6.4 and 6.6))
Main Definitions
Function limit (limx to c f(x))
Continuity at a point c
Continuity on a set
Derivative of a function at a point
Increasing, decreasing, strictly increasing, strictly decreasing function
Partition of an interval
Upper and lower sums U(f,P), L(f,P) of a function on a partition
Upper and lower integrals U(f) and L(f)
Riemann integrability
set of measure zero
Main Theorems
Sequence version of function limit
Limit laws for function limits
Sequence version of continuity
Function with multiple slots is continuous if individual slots are continuous (Example 5.3.6)
Inverse image under a continuous function of an open set is relatively open in the domain
Inverse image under a continuous function of a closed set is relatively closed in the domain
Image under a continuous function of a connected set is connected
Image under a continuous function of a compact set is compact
Extreme Value Theorem
Intermediate Value Theorem
If f is differentiable at c, then f is continuous at c
Derivative Rules (sum, difference, product, quotient, chain rules)
Fermat's Theorem: If f is differentiable at c and has a local max or min at c, then f '(c)=0
Rolle's Theorem
Mean Value Theorem
If f:[a,b] -> R is continuous and f ' > 0 on (a,b), then f is strictly increasing (plus variants
of this for strictly decreasing, increasing, decreasing)
L'Hopital's Rule
Lebesgue's Theorem: f is Riemann integrable on [a,b]
if and only if f is bounded and the set of discontinuities of f has measure zero.
A countable set has measure zero.
Any interval (a,b) with a < b does not have measure zero.
Riemann's condition
Properties of Integrals from class
If f is Riemann integrable on [a,b], and F(x) = intax f(t) dt, then F is
continuous on [a,b]
Fundamental Theorem of Calculus Part I: If f is continuous on [a,b],
and F(x) = intax f(t) dt, then F is differentiable on (a,b)
and F '(x)=f(x)
Fundamental Theorem of Calculus Part II: If f is continuous on [a,b], and F is
an antiderivative of f, then intab f(t) dt = F(b) - F(a).
To Math 317 home page
rmanning@haverford.edu