Skip to content

[Merged by Bors] - feat: Tendsto (√·) atTop atTop #33837

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Closed
Komyyy wants to merge 3 commits into from
Closed

[Merged by Bors] - feat: Tendsto (√·) atTop atTop #33837

Changes from all commits
Commits
Show all changes
3 commits
Select commit Hold shift + click to select a range
9f92a4d
feat: `Tendsto (√·) atTop atTop`
Komyyy Jan 11, 2026
c5d2841
chore: suggestions
Komyyy Jan 11, 2026
297758a
Merge branch 'master' into sqrt_at_top
Komyyy Jan 11, 2026
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
20 changes: 16 additions & 4 deletions Mathlib/Data/Real/Sqrt.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -119,12 +119,24 @@ variable {x y : ℝ}
theorem coe_sqrt {x : ℝ≥0} : (NNReal.sqrt x : ℝ) = √(x : ℝ) := by
rw [Real.sqrt, Real.toNNReal_coe]

@[continuity]
theorem continuous_sqrt : Continuous (√· : ℝ → ℝ) := by
@[continuity, fun_prop]
theorem continuous_sqrt : Continuous (√· : ℝ → ℝ) := by unfold sqrt; fun_prop

@[simp]
lemma map_sqrt_atTop : map (√·) atTop = atTop := by
unfold sqrt
exact NNReal.continuous_coe.comp <| NNReal.continuous_sqrt.comp continuous_real_toNNReal
change map (NNReal.toReal ∘ NNReal.sqrt ∘ Real.toNNReal) atTop = atTop
simp [← map_map]

@[simp]
lemma comap_sqrt_atTop : comap (√·) atTop = atTop := by
unfold sqrt
change comap (NNReal.toReal ∘ NNReal.sqrt ∘ Real.toNNReal) atTop = atTop
simp [← comap_comap]

lemma tendsto_sqrt_atTop : Tendsto (√·) atTop atTop := map_sqrt_atTop.le

theorem sqrt_eq_zero_of_nonpos (h : x ≤ 0) : sqrt x = 0 := by simp [sqrt, Real.toNNReal_eq_zero.2 h]
theorem sqrt_eq_zero_of_nonpos (h : x ≤ 0) : √x = 0 := by simp [sqrt, Real.toNNReal_eq_zero.2 h]

@[simp] theorem sqrt_nonneg (x : ℝ) : 0 ≤ √x := by
unfold sqrt
Expand Down