Proving C^k[a,b] is Not Closed in C^0[a,b]

[SP90]

Homework Statement

Is $C^{k}[a,b]$ closed in $C^{0}[a,b]?$

The Attempt at a Solution

$C^{k}[a,b]$ is obviously a subset of $C^{0}[a,b]$.

My gut feeling says no. I thought the best way would be to construct a function $f_{n}(x)$ which converges to $f(x)$ and where $f_{n}(x)$ is in $C^{k}[a,b]$ but $f(x)$ is not.

I thought maybe $f_{n}(x)=x^{k+1}sin(\frac{1}{nx})$ would do it since it's not k+1 differentiable at 0. But then $f(x)=0$ which can be differentiated infinitely (since each derivative is 0).

 

[clamtrox]

You are definitely on right track. Try writing your sequence as a series
$f_n = \sum_{i=0}^n a_n \sin(b_n x)$ and then choose a_n and b_n so that limit of f_n exists but f'_n diverges.
... Or just google "Weierstrass function", if you're lazy :)

 

[micromass]

If you're asking such questions, then you should always say which metric you're working with.