feat: holder continuity of suprema and infima

[CoolRmal]

This PR proves that pointwise suprema and infima of real-valued Holder continuous functions are Holder continuous, assuming the family is pointwise bounded above or below and the Holder constants are uniformly bounded.

The proof first shows the binary max/min statements, then uses induction for nonempty finite suprema/infima and the nonempty-finset convergence lemmas from #40578 together with the pointwise limit theorem for Holder continuous functions.

Created with the help of codex.

• feat(Topology/Order): convergence of suprema and infima #40578

[github-actions]

PR summary 69245d23a0

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ ciSup_eq_ciSup_finset
+ ciSup_eq_ciSup_finset_nonempty
+ instance [Nonempty ι] : IsDirectedOrder {F : Finset ι // F.Nonempty}
+ instance [Nonempty ι] : Nonempty {F : Finset ι // F.Nonempty}
+ tendsto_finset_inf'_ciInf
+ tendsto_finset_inf_ciInf
+ tendsto_finset_inf_iInf
+ tendsto_finset_sup'_ciSup
+ tendsto_finset_sup_ciSup
+ tendsto_finset_sup_iSup
++ ciInf
++ ciSup
++ finset_inf'
++ finset_sup'
++ inf
++ prodMk
++ sup
++ tendsto
- BddBelow.range_iInf_of_iUnion_range
- IsGLB.ciInf_eq
- IsGLB.ciInf_set_eq
- Set.Ici_ciSup
- WithBot.ciSup_empty
- WithBot.coe_iInf
- WithBot.coe_iSup
- WithBot.sSup_empty
- cbiInf_eq_ciInf_subtype
- cbiInf_id
- ciInf_and
- ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
- ciInf_image
- ciInf_inf_eq
- ciInf_le
- ciInf_le_of_le
- ciInf_mono
- ciInf_mono_of_forall_exists
- ciInf_prod
- ciInf_set_le
- ciInf_subtype
- ciInf_subtype_fun
- ciInf₂_le
- csInf_eq_of_forall_ge_of_forall_gt_exists_lt
- csInf_image
- exists_lt_of_ciInf_lt
- isGLB_ciInf
- isGLB_ciInf_set
- le_ciInf
- le_ciInf_exists
- le_ciInf_iff
- le_ciInf_set_iff
- map_ciInf
- map_ciInf_set
- map_csInf
- map_csInf'
- u_ciInf
- u_ciInf_set
- u_csInf
- u_csInf'

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 69245d2).

• +24 new declarations
• −2 removed declarations

+HolderOnWith.ciInf
+HolderOnWith.ciSup
+HolderOnWith.finset_inf'
+HolderOnWith.finset_sup'
+HolderOnWith.inf
+HolderOnWith.prodMk
+HolderOnWith.sup
+HolderOnWith.tendsto
+HolderWith.ciInf
+HolderWith.ciSup
+HolderWith.finset_inf'
+HolderWith.finset_sup'
+HolderWith.inf
+HolderWith.prodMk
+HolderWith.sup
+HolderWith.tendsto
-WithBot.ciSup_empty
+WithBot.iSup_empty
+ciInf_eq_ciInf_finset
+ciInf_exists_le
+ciSup_eq_ciSup_finset
-le_ciInf_exists
+tendsto_finset_inf_ciInf
+tendsto_finset_inf_iInf
+tendsto_finset_sup_ciSup
+tendsto_finset_sup_iSup

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 69245d23a0
Reference commit 9e7b1c1166

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[mathlib-bors]

This pull request is now in draft mode. No active bors state needed cleanup.

While this PR remains draft, bors will ignore commands on this PR. Mark it ready for review before using commands like bors r+ or bors try.