Assignment:
Let C_0 be the space of functions f:R --> R such that
lim f(x) = 0 as x goes to infinity and negative infinity
C_0 becomes a metric space with sup-norm
||f|| = sup { |f(x)| : x in R }
Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions from A has a convergent subsequence in the sup-norm.
Provide complete and step by step solution for the question and show calculations and use formulas.