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8 CLASS NOTES FOR APRIL 14, 2000<br />
The sequence xn c<strong>on</strong>verges to x, and the sequence 2 −n c<strong>on</strong>verges to 0. Since r > 0,<br />
this implies that we can find an integer n such that<br />
and<br />
In particular, this implies that<br />
d(xn, x) < r/2<br />
2 −n < r/2.<br />
B(xn, 2 −n ) ⊂ B(x, r).<br />
But since B(x, r) is c<strong>on</strong>tained in Vα, we have<br />
B(xn, 2 −n ) ⊂ Vα.<br />
This implies that B(xn, 2 −n ) can be covered by a finite number (in fact, just <strong>on</strong>e)<br />
of the Vα, which is a c<strong>on</strong>tradicti<strong>on</strong>. The proof is complete.
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