Fatou's Lemma and the Lebesgue's Convergence Theorem

For simplicity, we adopt the following rules: X denotes a non empty set, F denotes a sequence of partial functions from X into R with the same dom, s1, s2, s3 denote sequences of extended reals, x denotes an element of X, a, r denote extended real numbers, and n, m, k denote natural numbers. We now state several propositions: (1) If for every natural number n holds s2(n) ≤ s3(n), then inf rng s2 ≤ inf rng s3. (2) Suppose that for every natural number n holds s2(n) ≤ s3(n). Then (i) (the inferior real sequence of s2)(k) ≤ (the inferior real sequence of s3)(k), and (ii) (the superior real sequence of s2)(k) ≤ (the superior real sequence of s3)(k).