Pointwise Convergence in Metric Space (C[a,b],d_{\infty}) | Homework Solution

[Ted123]

Homework Statement

Suppose a sequence $(f_n)_{n\in\mathbb{N}}$ converges to a limit $f$ in the metric space $(C[a,b],d_{\infty})$ (continuous real valued functions on the interval [a,b] with the uniform metric.)

Show that $f_n$ also converges pointwise to $f$; that is for each $t\in [a,b]$ we have $f_n(t)\to f(t)$ in $\mathbb{R}$.

Homework Equations

Uniform metric: $$d_{\infty} (f,g) = \text{max}_{t\in [a,b]} |f(t)-g(t)|$$

The Attempt at a Solution

$f_n \to f$ in $(C[a,b],d_{\infty}) \iff d_{\infty}(f_n,f)\to 0$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff \text{max}_{t\in [a,b]} |f_n(t)-f(t)| \to 0$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff|f_n(t)-f(t)| \to 0$ for all $t\in [a,b]$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff f_n(t) \to f(t)$ for all $t\in [a,b]$

Does this prove it?

 

[Dick]

Looks ok to me.

 

[LCKurtz]

Homework Statement

Suppose a sequence $(f_n)_{n\in\mathbb{N}}$ converges to a limit $f$ in the metric space $(C[a,b],d_{\infty})$ (continuous real valued functions on the interval [a,b] with the uniform metric.)

Show that $f_n$ also converges pointwise to $f$; that is for each $t\in [a,b]$ we have $f_n(t)\to f(t)$ in $\mathbb{R}$.

Homework Equations

Uniform metric: $$d_{\infty} (f,g) = \text{max}_{t\in [a,b]} |f(t)-g(t)|$$

The Attempt at a Solution

$f_n \to f$ in $(C[a,b],d_{\infty}) \iff d_{\infty}(f_n,f)\to 0$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff \text{max}_{t\in [a,b]} |f_n(t)-f(t)| \to 0$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff|f_n(t)-f(t)| \to 0$ for all $t\in [a,b]$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff f_n(t) \to f(t)$ for all $t\in [a,b]$

Does this prove it?

You need to be careful what you are proving. You have all these $\iff$ implications which would lead one to believe that pointwise convergence and uniform convergence are the same. But they aren't. So look at your argument carefully and make sure the implications go in the direction to prove what you want to prove.

 

[Dick]

You need to be careful what you are proving. You have all these $\iff$ implications which would lead one to believe that pointwise convergence and uniform convergence are the same. But they aren't. So look at your argument carefully and make sure the implications go in the direction to prove what you want to prove.

Good point!

 

[Ted123]

Obviously to prove what I want I only need all steps to be $\implies$ but which step above is not "if and only if"? (is it the last step?)

 

[Dick]

|f_n(t)-f(t)|->0 for all t does not imply max |f_n(t)-f(t)|->0. Can you think of a counterexample?

 

[LCKurtz]

Obviously to prove what I want I only need all steps to be $\implies$ but which step above is not "if and only if"? (is it the last step?)

|f_n(t)-f(t)|->0 for all t does not imply max |f_n(t)-f(t)|->0. Can you think of a counterexample?

And I would add that if I were handing in a proof, I would use a tighter argument. While your implications in one direction are OK, you wouldn't have made that mistake if your argument went something like:

$$0 \le |f_n(t)-f(t)| \le ... \rightarrow 0$$

where you fill in the dots with reasons for each step.