n En=0m! I)Use Cauchy criterion prove that the sequence does not converge uniformly
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Solved in 2 steps
- Use Monotone Convergence Theorem to prove that (Sn) converges and find its limit.Let (hn),(tn) be sequences of bounded functions on A that converges uniformly on A to h,t respectively.show that (hntn) converges uniformly on A to ht(b) Give an example to show that the product (fngn) may not converge uniformly.
- Find the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly for x ∈ (0, +inf))? If yes, what is the uniform norm of fn(x)-f(x) on (0, +inf)?Does it converge or diverge? I do not understand how to find out.Does the Bounded Convergence Theorem hold if m(E)<∞ but we drop the assumption that the sequence {|fn|} is uniformly bounded on E?