Midterm 1

数学分析原理代写 Throughout this exam, you may use any theorems from class, class assignments, or in the class textbook Chapters 1-8

Throughout this exam, you may use any theorems from class, class assignments, or in the class textbook Chapters 1-8, for which a proof has been provided.

Question 1: Uniform Convergence, Part I  数学分析原理代写

Consider the sequence of functions, {f_n}_n, for n ∈ N₊, given by:

1. Findthe point-wise limit, f (x) = lim，
2. Prove that convergence to f (x) is notuniform over the interval (0, 1).
3. Prove the convergence to f (x) isuniform over the interval (1, ∞).

Question 2: Uniform Convergence, Part II  数学分析原理代写

Consider a function f : (0, ∞) → R, defined via the series:

1. Considering the Riemann integral ” 1 f (x)dx, prove the following limitexists:

(Hint: Show e! 1 f (x)dx exists.)

2. Prove that  f (x)  is  differentiable.

Question 3: Stone – Weierstrass 数学分析原理代写

We denote by C([0, 1]), the space of real valued continuous functions with domain [0, 1] ⊂ R. We also give C([0, 1]) a metric space structure with the supremum norm:

d(f, g) := sup{|f (x) − g(x)| x ∈ [0, 1]}  for  f, g ∈ C([0, 1]).

Now consider the subset A ⊂ C([0, 1]):

1. Prove that A is a dense subset ofC([0, 1]).
2. Assume f ∈ C([0, 1])satisfies:

Prove f = 0.