Sequences of Metric Spaces and an Abstract Intermediate Value Theorem

The notation and terminology used here are introduced in the following papers: [21], [22], [23], [3], [4], [2], [12], [18], [6], [1], [20], [7], [5], [8], [16], [14], [13], [15], [11], [19], [17], [9], and [10]. The following propositions are true: (1) Let R be a non empty subset of R and r0 be a real number. If for every real number r such that r ∈ R holds r ¬ r0, then supR ¬ r0. (2) Let X be a non empty metric space, S be a sequence of X, and F be a subset of Xtop. Suppose S is convergent and for every natural number n holds S(n) ∈ F and F is closed. Then limS ∈ F. (3) Let X, Y be non empty metric spaces, f be a map from Xtop into Ytop, and S be a sequence of X. Then f · S is a sequence of Y . (4) Let X, Y be non empty metric spaces, f be a map from Xtop into Ytop, S be a sequence of X, and T be a sequence of Y . If S is convergent and T = f · S and f is continuous, then T is convergent.