That's a good start. Here's the next step. For each i=1,2,3,..., choose with so that . (If the scalars are real then will be ±1. If they are complex then it will be some number on the unit circle.)

Now let . Then (c_0 norm), and . But . Therefore . Taking the limit of this as N→∞, you see that the sequence {a_i} is in , with .

On the other hand, if , then , from which you should be able to deduce that