to_archimedean — Mathlib

English

Another form of Jensen's inequality for sums of powers under nonnegativity: (same structure as Power Jensen).

Русский

Другая форма неравенства Дженса для сумм степеней при неотрицательности: та же структура, что и Power Jensen.

LaTeX

$$$\frac{\left(\sum_{i \in s} f(i)\right)^{n+1}}{|s|^n} \le \sum_{i \in s} f(i)^{n+1}$$$

Lean4

/-- Any conditionally complete linearly ordered field is archimedean. -/
instance (priority := 100) to_archimedean [ConditionallyCompleteLinearOrderedField α] : Archimedean α :=
  archimedean_iff_nat_lt.2
    (by
      by_contra! h
      obtain ⟨x, h⟩ := h
      have :=
        csSup_le (range_nonempty Nat.cast)
          (forall_mem_range.2 fun m =>
            le_sub_iff_add_le.2 <| le_csSup ⟨x, forall_mem_range.2 h⟩ ⟨m + 1, Nat.cast_succ m⟩)
      linarith)

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