Is Pointwise Convergent Sequences?

Pointwise convergence defines the convergence of functions in terms of the conver- gence of their values at each point of their domain. Definition 5.1. Suppose that (fn) is a sequence of functions fn : A → R and f : A → R. Then fn → f pointwise on A if fn(x) → f(x) as n → ∞ for every x ∈ A.

Does pointwise convergence imply convergence?

Uniform convergence implies pointwise convergence, but not the other way around. For example, the sequence fn(x)=xn from the previous example converges pointwise on the interval [0,1], but it does not converge uniformly on this interval.

How do you show a sequence converges pointwise?

Consider the sequence {fn} of functions defined by fn(x) = sin(nx + 3) √ n + 1 for all x in R. fn(x) = 0 for all x in R. Therefore, {fn} converges pointwise to the function f ≡ 0 on R.