6: Sequences and Series of Functions, Convergence

Proposition 1.2. Let (X, d) be a metric space. Let (x)j=k be a sequence of elements of X. Let x, x′ be elements of X. Assume that the sequence (x)j=k converges to x with respect to d. Assume also that the sequence (x)j=k converges to x ′ with respect to d. Then x = x′. Proposition 1.3. Let a < b be real numbers, and let f : [a, b] → R be a function which is both continuous and strictly monotone increasing. Then f is a bijection from [a, b] to [f(a), f(b)], and the inverse function f−1 : [f(a), f(b)]→ [a, b] is also continuous and strictly monotone increasing.