You don't need to consider L^2 norms. Just look at coordinates. If you have a nested sequence of bounded closed sets in R^n then (for j=1,2,...,n) their j'th coordinates form a nested sequence of bounded closed sets in R. By the nested interval theorem for the real line, this sequence has a nonempty intersection. If x_j is a point in this intersection then the point

is in the intersection of the sets in the R^n sequence.

If you put an extra condition on the sets in R^n, such as asking that their diameters should tend to zero, then their intersection will be a single point.