upper boundlower boundbounded if it has both lower and upper bound
maximum upper bound and minimum lower bound and supremum, infimum,
Axiom of Completeness a nonempty set of real numbers has an or if it is bounded below or above
Archimedean Property,
Nested Interval Property
Sequences
converges if , or bounded if Cauchy if monotone if
Monotone Convergence a monotone bounded sequence is convergent
Bolzano-Weierstrass a bounded sequence has a convergent subsequence
Topology
Let
limit point, isolated point is not a limit point
openclosed is open, contains all its limit points
compactHeine-Borel Theorem
is bounded and closed
Every open cover of has a finite subcover
closure the union of with its limit points
connectedperfect every point of is a limit point.
Union of arbitrarily many open sets is open.
Union of finitely many closed sets is closed.
Intersection of arbitrarily many closed sets is closed.
Intersection of finitely many open sets is open.
Functions
Let
continuous
uniformly continuous
monotonecomposition for continuous, is continuous
compactness compact then compact.
connectedness connected then connected.
Extreme Value Theorem
Intermediate Value Theorem, .
Derivative
Differentiability Let . For , the derivative of at is . If this exists, then is differentiable at . If exists for all , then is differentiable on .
If is differentiable at , then it is continuous at .
Sequential Criterion differentiable at if and only if , implies converges
has a relative extrema at if where implies is an upper or lower bound for
If exists and is a relative extrema, then
Interior Extremum Theorem Let be differentiable on . If has max or min value at , then .
Darboux Theorem If differentiable on and , then there is where .
Rolle’s Theorem Let differentiable on . If , then there is a point where .
Mean Value Theorem If is continuous on and differentiable on , then there exists such that
Generalized Mean Value Theorem If and are continuous, then