Uniform Boundedness Principle

The following two propositions are true: (1) For every sequence s1 of real numbers and for every real number r such that s1 is bounded and 0 ≤ r holds lim inf(r s1) = r · lim inf s1. (2) For every sequence s1 of real numbers and for every real number r such that s1 is bounded and 0 ≤ r holds lim sup(r s1) = r · lim sup s1. Let X be a real Banach space. One can verify that MetricSpaceNormX is complete. Let X be a real Banach space, let x0 be a point of X, and let r be a real number. The functor Ball(x0, r) yielding a subset of X is defined as follows: (Def. 1) Ball(x0, r) = {x;x ranges over points of X: ‖x0 − x‖ < r}. The following propositions are true: (3) Let X be a real Banach space and Y be a sequence of subsets of X. Suppose ⋃ rng Y = the carrier of X and for every element n of N holds Y (n) is closed. Then there exists an element n0 of N and there exists