Lectures

Question 1

fn(x) = 1/n sin(nx). Each fn is differentiable. Show that the lim = 0 for all x in R. Show that Lim fn’(x) does not exist at x = π.

Answer 1

Lim fn(x) = Lim n -> ∞ [1/n sin(nx)]
Equals:
Lim n -> ∞ [-1/n] <= Lim n -> ∞ [1/n sin(nx)] <= Lim n -> ∞ -1/n by the squeeze theorem.
0 <= ? <= 0. Proves it is 0.

Lim cos( πx) only exists for x in 2πZ. For instance cos( πn) = Lim n -> ∞ (-1)^n doesn’t exist.

Question 2

What is pointwise convergence?

Answer 2

Let be a sequence of functions, fn:x -> y any topological spaces x & y. Then f is the "pointwise limit" of   
if f(x) =  Lim n -> ∞ fn(x) for all x in X.

It is essentially that given any x, the function equates to lim fn(x).

Question 3

What is Uniform Convergence?

Answer 3

Let be a sequence of functions fn: x -> y metric space. Then f is the uniform limit of if f is the Limit of using the distance: sup (dist(g, h) = sup{dist(g(x), h(x))} for x in.
In other words, for each ɛ > 0, there is a N such that
sup{ dist{ fn(x), f(x) } | x in X} < ɛ for all n > N.
<=> dist( fn(x), f(x) ) < ɛ for all n > N and x in X.

Question 4

fn(x) = x ^n fn -> f pointwise where f(x) = { 0 if x in [0,1)
1 if x = 1 }
fn:[0,1] -> R

What is the uniform limit?

Answer 4

There is no uniform metric because it does not converge in sup-metric or sup-norm.