Square root function and the logarithm function
the Zariski-closed sets
Geometric view of complex functions
- tips: proof techniques, definitions, definitions – links – proof, notations (quantifiers),
- Cauchy Sequence: Introduction
- limit of subsequence = limit of sequence
- Fibonacci: limit of ratio of two successive numbers is golden ratio with proof, find the nth number
- triangle inequality: real numbers, complex numbers
- limit of functions
- Sequential limits in functions
- Topology:Made easy, introduction Why 1, why 2, riddles, 生活中的拓扑，
- Topology Sorting: class choosing,
Epsilon Definition of Supremum and Infimum – a rigorous proof, how to use definitions to prove.
- Why sequence infima (suprema) are increasing (decreasing)?
- Infimum/Supremum
q1:
(5) The infimum and supremum are related via \text{inf } S = -\text{sup } (-S), where -S = \{ -s \colon s \in S \}. This is often convenient for proofs.
q2: The proofs of the two statements are more or less identical (and can be formally translated to each other by remark (5) above). Here is the proof of the first statement. If x = \text{inf } S, then x+\epsilon cannot be a lower bound for S, so there must be an element of S that is bigger than it. On the other hand, if x is a lower bound that is not the infimum, then there is a larger lower bound x’ for S. Let \epsilon = x’-x; then there is no s \in S such that s< x+\epsilon = x’.\ _\square
q3: Note that this property is not true for the rational numbers: the set of all rational numbers less than \sqrt{2} has an upper bound that is rational (e.g. 2), but there is no least rational upper bound \big(there are rational numbers less than \sqrt{2} + \epsilon for any \epsilon > 0\big).
- Limit superior and limit inferior
- Twin Prim conjecture