This is a fairly long and tedious proof. Here's an outline.

Let be the standard basis for , where has a 1 in the n'th coordinate position and zeros everywhere else. Write for a general element of .

Given , show that the map sending to is a bounded linear functional on , and that its norm as a linear functional is equal to the norm of .

To complete the proof, you need to show that every bounded linear functional on is of that form. So let . For n=1,2,3,... let . Show that the element defined in this way is in , and that the functional defined by as in the previous paragraph is equal to .