Math 2.0

statement theorem

Existence of the infimum

Statement

Let SS be a nonempty subset of the real numbers that is bounded below. Then SS has an infimum in R\mathbb{R}.

Remarks

Proof: the set −S={−s:s∈S}-S = \{-s : s \in S\} is nonempty and bounded above. By the completeness axiom, it has a supremum bb. Then −b-b is the infimum of SS. Another proof: the set of lower bounds of SS is nonempty and bounded above by each element of SS, and its supremum is inf⁡S\inf S.

Explanations

L1 human

If a set of real numbers has a floor, then it has a highest floor. This follows from the completeness axiom by a reflection of the number line through 00, which turns floors into ceilings.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 6 nodes in this order. The foundations come first, and the list ends with this node.

Real numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionInfimum, definition, no evidence levelInfimumdefinitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of the infimum, statement, no evidence levelExistence of the infi…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Real numbers object
  2. Set bounded above definition
  3. Infimum definition
  4. Supremum definition
  5. Completeness axiom statement
  6. Existence of the infimum statement

Sources

  1. Web https://en.wikipedia.org/wiki/Least-upper-bound_property, Statement for real numbers

What links here

No node links to this node yet.