Answer:
The statement "aₙ converges to L ∈ X if and only if every subsequence of (aₙ)ₙ₌₁^∞ has a sub-subsequence which converges to L" can be formalized as follows:
Definition 1: Convergence of a sequence
A sequence (aₙ)ₙ₌₁^∞ in a metric space (X, d) is said to converge to a point L ∈ X if for every positive real number ɛ > 0, there exists a positive integer N such that for all n ≥ N, the distance between aₙ and L, denoted as d(aₙ, L), is less than ɛ.
Definition 2: Subsequence
A subsequence of a sequence (aₙ)ₙ₌₁^∞ is obtained by selecting an infinite number of terms from the original sequence in their given order.
Definition 3: Sub-subsequence
A sub-subsequence of a subsequence (bₙₖ)ₖ₌₁^∞ is obtained by selecting an infinite number of terms from the subsequence in their given order.
Definition 4: Convergence of a sub-subsequence
A sub-subsequence (bₙₖₗ)ₗ₌₁^∞ of a subsequence (bₙₖ)ₖ₌₁^∞ is said to converge to a point L ∈ X if for every positive real number ɛ > 0, there exists a positive integer N such that for all l ≥ N, the distance between bₙₖₗ and L, denoted as d(bₙₖₗ, L), is less than ɛ.
The statement can now be formally stated as:
Statement: A sequence (aₙ)ₙ₌₁^∞ in a metric space (X, d) converges to a point L ∈ X if and only if every subsequence of (aₙ)ₙ₌₁^∞ has a sub-subsequence that converges to L.
In simpler terms, this statement says that a sequence converges to a certain point if and only if every subsequence of that sequence contains a further subsequence that converges to the same point.