Real Analysis Answers

Prove that a sequence in a metric space cannot converge more than one limit

Let (x,d) be a metric space and A a subset of X .prove that A is closed if and only if ,if each convergent sequence of points of A converges to a point of A

Let f and g be continuous functions from a metric space X to metric space Y such that they agree on a dense subset of a X .then prove that f=g

Prove that any uniformly continuous function from a dense subset of a metric space X to a complete metric space Y has a uniformly continuous function from X to Y

For x and y in R, let d(x,y)=(|x-y|)/(1+|x-y|),prove that d defines a.bounded metric on R

Prove that a uniformly continuous function from metric space to another metric space takes cauchy to cauchy sequences

Test the uniform convergence of the series

Σ∞

𝑛=1 x[𝑛/(1+𝑛^2𝑥^2)−(𝑛+1)/(1+(𝑛+1)^2𝑥^2)]

Let Y be a subspace of a metric space (X, d). Then show that 𝐹⊆𝑌 is closed in Y if and only if

𝐹=𝑌∩𝐻 for some closed subset H of X.

If 𝑓 is a continuous function from a metric space X into a metric space Y and {𝑥𝑛} is a sequence in X

which converges to 𝑥 then show that the sequence {𝑓(𝑥𝑛)} converges to 𝑓(𝑥) in Y.

A function is called “piecewise linear” if it is (i) continuous and (ii) its graph consists of finitely many linear segments. Prove that a continuous function on an interval [a, b] is the uniform limit of a sequence of piecewise linear functions.