Question about open sets?

Let A be the set of all sequences {s_n} (n=1 to infinity) in L^2 (L^2, being the typical definition, = {{s_n} (n=1 to infinity) : the sum n=1 to infinity of s_n^2 is finite} such that the sum of s_n^2 is less than 1. Prove that A is an open subset of L^2.

I figure that I am using this definition to prove this....

Let M be a metric space. We say the subset A of M is open of M if for all x in A there exists r>0 such that the entire open ball B(x;r) is contained in A.

B(x,r) = {a in A: rho(x,a)<r}

Since we are told we are in L^2, I'm trying to figure out how to use that....

I had this question answered earlier, but today my teacher told us that we should be using a particular theorem (shown below) to prove this..somehow.

The norm for sequences in L^2 has the following properties:

1) ||s||_2 => 0, s is in L^2

2) ||s||_2 =0 iff s={0}(n=1 to infinity)

3) ||cs||_2 = |c|||s||_2 where c is a constant

4) ||s+t||_2 <= ||s||_2 + ||t||_2

Any ideas?