Inns) Consider the sequence { snf = { 1 } in the metric Space R. Prove...
Q3. lut (H, J) be a compart metric space . Suppose ful is sequence of continuous function fo: M-IR which converges uniformly to f: MIR Prove that the sequence 4 Converges uniformly to f².
In the following exercises, consider the metric space R with the discrete metric and the subset A [0, 1 C R. 15) True/False: A is closed. 16) True/False: A is open 17) True/False: Every point of A is a limit point of A. 18) Calculate the boundary of A. 19) True/False: For all Xo E X and all ε > 0, if B(x0, e) contains a point of A besides xo, then A C B(xo, e)
In the following exercises,...
Theorem 8.4. A sequence of points in a metric space has at most one limit. Proof. We will show that a sequence of points in a metric space has at most one limit. We will do this by contradiction, by supposing that some sequence has two different limits, and deriving the contradiction 1<]. Suppose the sequence (Pk)'_, converges to two different limits a and b. Let ε = d (a,b). <This choice for ɛ will be explained by a calculation...
3. Suppose X is a metric space with a sequence of points Xn e X with the property that for each n + m we have d(Xn, Xm) = 1. Prove that no subsequence of xn converges, and that therefore X is not compact. Hint: You could use the previous problem.
Exercise 5 (based on Tao). Let (X,d) be an arbitrary metric space. Prove the following statements (1) If a sequence is convergent in X, all its subsequences are converging to the same limit as the original sequence. (2) If a subsequence of a Cauchy sequence is convergent, then the whole sequence is convergent to the same limit as the subsequence. (3) Suppose that (X,d) is complete and Y S X is closed in (X,d). Then the space (Y,dlyxy) is complete....
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R. Suppose that, for each the sequence (fe(x))ke N 1s a monotonic sequence which converges to (x). Show that r є X, k)kEN Converges to j uniformly.
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R....
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
5. Consider the metric space consisting of the set C([0, 1], R) - the set of all real valued, continuous functions on (0,1) - and the metric 1/2 P(5.9) = ([*(86) – 9()° dx) Demonstrate that this metric space is not complete.
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in class is a valid norm on R2 (b) Prove that in R2, any open ball in 12 ("Euclidean metric") can be enclosed in an open ball in the loo norm ("sup" norm). (c). Say I have a collection of functions f:I R. Say I (1,2). Consider the convergence of a sequence of functions fn (z) → f(x) in 12-Show that the convergence amounts to...