Sample midterm 2 questions-
1. Let (fn) be a sequence of uniformly continuous functions on an interval (a, b), and suppose that fn converges uniformly to a function f. Prove that f is uniformly continous on (a, b).
2. Prove that the functions
d1(x, y) = (x - y)4, d2(x, y) = 1 + |x - y|,.
![377_Figure.png](https://secure.tutorsglobe.com/CMSImages/377_Figure.png)
are not metrics on R.
3. Find the radius of convergence of the series
n=0∑∞xn/n√n, n=0∑∞4nx2n+1, n=0∑∞xn^2.
4. Suppose that fn converges uniformly to f on a set S ⊆ R, and that g is a bounded function on S. Prove that the multiplication g · fn converges uniformly to g · f.
5. Let (fn) be a sequence of bounded functions on a set S, and suppose that fn → f uniformly on S. Prove that f is a bounded function on S.
6. Let (fn) be a sequence of real-valued continuous functions defined on the interval [0, 1]. Suppose that fn converges uniformly to a function f. Define a global bound M according to
M = sup{|fn(x)|: n ∈ N, x ∈ [0, 1]}.
Prove that M is finite.
7. (a) Prove that the function
d(x, y) = min{|x - y|, 1}
is a metric on R.
(b) Is the set (-5, 5) open with respect to this metric? Prove your assertion.
8. (a) Find the radius of convergence of the power series
f1(x) = n=1∑∞xn/n2, f2(x) = n=0∑∞ x2n/2n.
(b) Show that the series
f3(y) = n=1∑∞(1/n2)(y/1 + y2)n
converges for all values of y ∈ R.
9. Consider the function defined on the domain [0, ∞) as
![689_Figure1.png](https://secure.tutorsglobe.com/CMSImages/689_Figure1.png)
Define a sequence of functions on the interval [0, 1] according to fn(x) = g(nx).
(a) What is the maximum value of g(x), and where is it attained?
(b) Sketch the functions f1(x), f2(x), and f3(x) on [0, 1].
(c) Prove that fn converges point-wise to a function f on [0, 1], and determine f.
(d) Does fn converge uniformly to f on [0, 1]? Prove your assertion.
10. Let (fn) be a sequence of continuous functions on [a, b] that converges uniformly to f on [a, b]. Show that if (xn) is a sequence in [a, b] and if xn → x, then limn→∞ fn(xn) = f(x).