Let f : D ⊆ R → R be uniformly continuous and {xn} be a Cauchy sequence in D. Then show that {f(xn)} is also Cauchy.
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Let f : D ⊆ R → R be uniformly continuous and {xn} be a Cauchy sequence in D.
Then show that {f(xn)} is also Cauchy.
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- If g : A → R is continuous and (an) is a Cauchy sequence in A, does it follow that the sequence (g(an)) is Cauchy?Let X be a Banach space an {xn} be a sequence in. X. Provethat if {xn} converges in norm in X, then it converges weakly to thesame limitDoes the Bounded Convergence Theorem hold if m(E)<∞ but we drop the assumption that the sequence {|fn|} is uniformly bounded on E?
- Prove that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is uniformly convergent on R.Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1].Find limits of the following sequences or prove that they are divergent.(a) an =√n(−1)^n
- 3. If g : A → R is continuous and (an) is a Cauchy sequence in A,does it follow that the sequence (g(an)) is Cauchy?Let fn(x) = x/(n^2+x^2) for x ∈ R. Show that the sequence {fn} converges uniformly to the function that is everywhere zero.Prove that the sequence {cn} converges to c if and only if the sequence {cn- c} converges to 0.
- Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise A sequence (xn) ⊆ X is called eventually constant if there is N ∈ N such that xm=xn; m, n > N. Show that any eventually constant sequence converges.let y=f(X) be a uniformly continuous function on a set I. then for a given sequence (a_n)_{n∈N} ⊆ I , which of the following statement or statements is/are correct ? a-if (a_n) is Cauchy, then (f(a_n)) is also a Cauchy sequence. b- if (a_n) is convergent, then (f(a_n)) is also a Convergent sequence. c- y=f(X) is continuous at every point of the set I. d-all of the above. e- none of the above.Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.