Uniform Convergence Proof

--- Prove: If fn converges to f uniformly and gn converges to g uniformly, then converges uniformly to f/g. --- See attached file for full problem description.

Uniform Convergence of arctan(nx)

Discuss the uniform convergence of the sequence of functions (see the attached file for the function)

Uniform Convergence

(See attached file for full problem description and equations) --- Prove the following theorem. Let be continuous functions on a closed bounded interval . Then uniformly on if and only if for every such that . ---

Find example of uniformly convergent series/Weierstrass test

(See attached file for full problem description and equations) --- Find an example of a sequence of continuous functions on on [0,1] such that the series converges uniformly on [0,1] but the series diverges. Is it a counterexample for the Weierstrass test? ---

Uniform Convergence of series

(See attached file for full problem description and equations) --- Prove the following theorem: If for all and every , and the series converges uniformly in [a,b], then converges uniformly in [a,b]. ---