Sequences of Metric Spaces and an Abstract Intermediate Value Theorem 1

Relations of convergence of real sequences and convergence of metric spaces are investigated. An abstract intermediate value theorem for two closed sets in the range is presented. At the end, it is proven that an arc connecting the west minimal point and the east maximal point in a simple closed curve must be identical to the upper arc or lower arc of the closed curve. provide the notation and terminology for this paper. One can prove the following propositions: (1) Let R be a non empty subset of R and r 0 be a real number. If for every real number r such that r ∈ R holds r ≤ r 0 , then sup R ≤ r 0. (2) Let X be a non empty metric space, S be a sequence of X, and F be a subset of X top. Suppose S is convergent and for every natural number n holds S(n) ∈ F and F is closed. Then lim S ∈ F. (3) Let X, Y be non empty metric spaces, f be a map from X top into Y top , 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 X top into Y top , 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. (5) For every non empty metric space X holds every function from N into the carrier of X is a sequence of X. (6) Let s be a sequence of real numbers and S be a sequence of the metric space of real numbers such that s = S. Then (i) s is convergent iff S is convergent, and (ii) if s is convergent, then lim s = lim S. (7) Let a, b be real numbers and s be a sequence of real numbers. If rng s ⊆ [a, b], then s is a sequence of [a, b] M .