isCauSeq_iff_cauchySeq

theorem isCauSeq_iff_cauchySeq : ∀ {α : Type u} [inst : NormedField α] {u : ℕ → α}, IsCauSeq norm u ↔ CauchySeq u := by sorry

This theorem states that in a normed field, a sequence is a Cauchy sequence according to the `IsCauSeq` definition if and only if it is a Cauchy sequence according to the `CauchySeq` definition. Here, a normed field is a field equipped with a norm, which is a function that assigns a non-negative size or length to each element. The `IsCauSeq` definition says that a sequence is Cauchy if the distance between its entries, measured by a given function, tends to zero. The `CauchySeq` definition says that a sequence is Cauchy if it is Cauchy at infinity in a uniform space, that is, if its values become arbitrarily close to each other as one moves far enough along the sequence. The theorem demonstrates the equivalence of these two definitions in the context of normed fields.

More concisely: In a normed field, a sequence is a Cauchy sequence according to the `IsCauSeq` definition if and only if it is a Cauchy sequence according to the `CauchySeq` definition.