Sample midterm 2 questions find the radius of convergence

Question

Sample midterm 2 questions-

1. Let (f_n) be a sequence of uniformly continuous functions on an interval (a, b), and suppose that f_n 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|,

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∑^∞x^{n^2}.

4. Suppose that f_n converges uniformly to f on a set S ⊆ R, and that g is a bounded function on S. Prove that the multiplication g · f_n converges uniformly to g · f.

5. Let (f_n) be a sequence of bounded functions on a set S, and suppose that f_n → f uniformly on S. Prove that f is a bounded function on S.

6. Let (f_n) be a sequence of real-valued continuous functions defined on the interval [0, 1]. Suppose that f_n 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.