Math 2.0

Real functions Theorem

Sequential characterization of limits

Builds on Limit of a functionLimit of a sequence

Statement

Let D⊆ℝD \subseteq \mathbb{R}, let f ⁣:D→ℝf \colon D \to \mathbb{R}, let aa be a limit point of DD, and let LL be a real number. Then lim⁡x→af(x)=L\lim_{x \to a} f(x) = L if and only if the following holds. For each sequence (xn)(x_n) in D∖{a}D \setminus \{a\} with xn→ax_n \to a, f(xn)→Lf(x_n) \to L.

Remarks

This turns facts about limits of sequences into facts about limits of functions. Examples are the limit laws for sums, products, and quotients. It is also a tool to show that a limit does not exist. Find two sequences tending to aa whose images have different limits. Mathlib states a general form as tendsto_nhds_iff_seq_tendsto.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSequential characterization of limits, statement, no evidence levelSequential characteri…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Limit of a sequence definition
  6. Sequential characterization of limits statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Limit_of_a_function, In terms of sequences
About this page
Tag000MXHEY22
URIm2://000MXHEY22
Aliasesheine-definition-of-limit
Kindstatement, theorem
WritingW0 stub
MSC26A03
StatusThis copy of the site has no build data. Run m2 status 000MXHEY22.
UsesLimit of a function, Limit of a sequence

Markdown · JSON · Edit on GitHub

What this page 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 page is. They do not measure its truth.

What links here

Used by