During the lecture of Thursday 11/9, we started Chapter 6 Sequences and Series of Functions. We first learned the definitions of pointwise convergence and uniform convergence. Then we worked on the template of proving fn → f is uniform on domain A and fn is not convergence uniform on domain A. After learning the definitions, we applied several examples and discussed about if those sequence of functions have pointwise limit or if they convergence uniformly on domains.

Proof Template for Pointwise Convergence: Let x ∈ A be arbitrary (A is the domain of fn(x)), [WTS: (fn(x)) → f(x)]. Let ε > 0 be arbitrary… Choose N be a natural number (note N may be depend on x), consider n ≥ N…, then |fn(x) – f(x)| < ε. Thus (fn(x)) → f(x), but x ∈ A was arbitrary which implies fn → f pointwise on A.

Prove fn → f uniformly on A: Let ε > 0 be arbitrary… Choose N be a natural number. Let n ≥ N and x ∈ A be arbitrary… then |fn(x) – f(x)| < ε.

Prove fn does not convergence uniformly on A: ∃ε。 > 0 such that ∀ natural number N, ∃n ≥ N and x ∈ A such that |fn(x) – f(x)| ≥ ε。.