Tendsto res To Imag Axis of tendsto at Im Infty

lemma tendsto_resToImagAxis_of_tendsto_atImInfty {F : ℍ → ℂ} {c : ℂ}    (hF : Tendsto F atImInfty (nhds c)) :    Tendsto F.resToImagAxis atTop (nhds c) := by  rw [Metric.tendsto_atTop]  intro ε hε  -- Get eventual proximity from hF  have hF_met : ∀ᶠ z in atImInfty, dist (F z) c < ε := Metric.tendsto_nhds.mp hF ε hε  obtain ⟨A, hA⟩ := Filter.eventually_atImInfty.mp hF_met  use max A 1  intro t ht  have ht_pos : 0 < t := lt_of_lt_of_le one_pos (le_of_max_le_right ht)  simp only [Function.resToImagAxis, ResToImagAxis, ht_pos, ↓reduceDIte]  set z : ℍ := ⟨Complex.I * t, by simp [ht_pos]⟩  have hz_im : z.im = t := by simp [UpperHalfPlane.im, z]  exact hA z (by simpa [hz_im] using le_of_max_le_left ht)