I am trying to include the fact that $\lim_{N\to\infty} \sum_{n=0}^{N} x^n=\frac{1}{1-x}$ for $|x|<1$.
Perhaps I could use $C[0,1]$ as a subspace of $C(\mathbb{R})$(or some bigger space?), and then view $\{x_N\}=\{\sum_{n=0}^{N} x^n\}$ as example of a sequence which has it's limit not in $C[0,1]$, but where all of the terms are?
Is this thinking correct?
Thank you.
As discussed in comments, it is not clear what exactly you want an example of, but I think it sounds like you are seeking a vector space $V$ over a field $K$, with a subset $U \subset V$ such that $U$ fails to be a vector space?
If so, let $V$ be $\mathbb{R}^2$ with the usual vector space structure over $\mathbb{R}$.
Take $U$ to be the upper half plane, $$U = \{\, (x,y) \, | \, y \geq 0 \, \}.$$ $U$ fails to be a subspace for it fails closure when vectors are acted on by scalars. This can be easily checked by seeing, $(-1) \cdot (1,1) = (-1,-1) \not\in U$, for example.
For failing closure of addition of vectors, take $U$ to be the subset that is quadrant 1 union quadrant 3, $$U = \{ \, (x,y) \, | \, x, y \geq 0 \text{ or } x, y\leq 0 \, \}.$$ The vectors $(-1,0)$ and $(0,1)$ are in $U$ but $(-1,0) + (0,1) = (-1,1)$ is not.