代做MATH 451 Fall 2019 Practice Final代写Web开发

MATH 451 Fall 2019

Practice Final

Problem 1:

Let S, T be non-empty subsets of R with the property that s < t for all s ∈ S and all t ∈ T. Prove that sup S and inf T exist and satisfy the relation sup S ≤ inf T. It it also true that sup S < inf T? Give a proof or a counterexample.

Problem 2:

Compute the limit of the following sequence

Problem 3:

Prove that the function  defined on (0, ∞) is uniformly continuous.

Problem 4:

Show that for all x ≥ 0 we have sin(x) ≤ x.

Problem 5:

Let f : [a, b] → R be a function. Assume that there exists a sequence (hn) of step functions hn : [a, b] → R converging uniformly to f. Show that f is integrable, without appealing to the theorem about uniform. convergence and integrability proved in class.