Tends-to relation

Mathematical statement

The relation x→a is directional language used in limits. By itself it does not say whether function values converge, nor does it require x eventually to equal a.

Representative formula (exact TeX)

x\to a

Assumptions

• A mathematical structure supporting neighborhoods and approach, such as a metric or topology, has been specified.

Symbol scope

• x — varying expression or variable
• a — target of approach
• → — tends-to relation

Intuition, definition, and misconceptions

Approach describes a process or relation rather than a single substitution. The value x may come arbitrarily close to a without being equal to a at every stage under discussion.

A complete function-limit statement must also specify the target of f(x), the order of quantifiers, and neighborhood conditions. This object supplies only the approach component.

Mathlib expresses convergence of a function relative to source and target filters as Filter.Tendsto f l₁ l₂, which is definitionally equivalent to Filter.map f l₁≤l₂.

A common mistake is to read x→a as x=a, or to claim that a bare arrow already equals a complete Filter.Tendsto or epsilon-delta definition before source and target filters are specified.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Function mapping — A function limit studies the behavior of mapped values as inputs approach a target.

Verification boundary

Lean formalizes the core filter semantics of Filter.Tendsto. The bare notation x→a still needs a function and source and target filters, and this evidence does not prove a limit for any particular function.

formalized_filter_tendsto_core

• evidence-sha256-38b848281997cbb1a2c1ea5559c6937dbb67ee03ac0be338756320a99e0f8b1d

Next obligation

Create objects for commonly used source and target filters such as atTop, nhds, and nhdsWithin, then connect them separately to sequence limits and pointwise function limits.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.