When proving a set, A, is not closed in (C[0,1], ||.||) how does one go about constructing a useful sequence of functions in A that you know will not converge to a point in A. Whenever I look at solutions to these kinds of problems, they make sense in hindsight but never come to mind when I'm attempting them myself. Any advice? Common things to look for or common functions used for certain problems?
An example would be the set A = { f belongs to C[0,1] and f(0)=0 } In the space (C[0,1], ||.||_1) this was shown to be not closed by considering the functions connecting the points (0,0), (1/n, 1) and (1,1) with straight lines.
A discussion of various examples would be great as I can hopefully gain some exposure to the methods used to solve these types of problems. Thanks in advance!
You can start by looking at $A^c=\{f\in C([0,1]):f\notin A\}$ and taking an example of a $f\in A^c$, then ask yourself how can you approximate $f$ with elements of $A$. There isn't a general method, since the construction of a sequence $f_n\in A$ converging to some $f\notin A$, depends on the definition of $A$ itself. One suggestion I can give is to keep it simple, when looking for $f\notin A$, start with a simple example, and so for $f_n\in A$.
For example, to show that $A=\{f\in C([0,1]):f(0)\neq0\}$ is not closed, you can take $0\in A^c=\{g\in C([0,1]):g(0)=0\}$ and approximate it with the sequence $f_n(x)=\frac{1}{n}$.
For the subset $A=C^1([0,1])\subseteq C([0,1])$ (the set of differentiable functions $[0,1]\rightarrow\mathbb R$) consider a discontinuous $f:[0,1]\rightarrow\mathbb R$ having an angle on a point $x=p$ (like $x\mapsto|x|$ at $x=0$) and approximate it with function which are "smoother" on $p$. For example, $$|x|:[-1,1]\rightarrow\mathbb R$$ can be approximated by the sequence $$f_n:x\mapsto\sqrt{x^2+\frac{1}{n}}$$and each $f_n$ is differentiable.
Another note: You say that "[$\{f\in C([0,1]):f(0)=0\}$] was shown to be closed by considering the functions connecting the points $(0,0), (1/n, 1)$ and $(1,1)$ with straight lines". When you're proving that a set $X$is closed, you can't take a single sequence and prove that is converges to an element $\in X$ (notice that the sequence you mention is not even convergent in $C([0,1])$), you need to show that every convergent sequence in $X$ converges to an element of $X$.