Such as can I construct a sequence by reversing the order of the approximating sequence of $\frac{1}{3}$? So such inverse would look like $\left\{….,0.333333333,0.3333333,0.333333,...0.3\right\}$.
I had this question when I was constructing a sequence that is bounded between 0 and 1/2 and not convergent to 0 for a homework question.
A sequence is ultimately a map whose domain is $\mathbb N$, but the object that you describe (a) has no first term, and (b) terminates.
If you want, I suppose you could define an object whose indices run from $-\infty$ to $0$, but that's not appreciably different from looking at the original sequence while standing on your head.