1772 CHAPTER 56. APPROXIMATION BY RATIONAL FUNCTIONS

Now from the choice of b1, the series

∞

∑k=0

(b1−bz−b

)k

=1(

1− b1−bz−b

)converges absolutely independent of the choice of z ∈ K because∣∣∣∣∣

(b1−bz−b

)k∣∣∣∣∣< 1

2k .

By Corollary 56.1.7 the same is true of the series for 1(1− b1−b

z−b

)n . Thus a suitable partial sum

can be made uniformly on K as close as desired to 1(z−b1)

n . This shows that b satisfies P

whenever b is close enough to b1 verifying that S is open.Next it is shown S is closed in V. Let bn ∈ S and suppose bn→ b∈V. Then since bn ∈ S,

there exists a rational function, Qbn such that

||Qbn −R||K,∞ <ε

2.

Then for all n large enough,12

dist(b,K)≥ |bn−b|

and so for all n large enough, ∣∣∣∣b−bn

z−bn

∣∣∣∣< 12,

for all z ∈ K. Pick such a bn. As before, it suffices to assume Qbn , is of the form 1(z−bn)

n .

ThenQbn (z) =

1(z−bn)

n =1

(z−b)n(

1− bn−bz−b

)n

and because of the estimate, there exists M such that for all z ∈ K∣∣∣∣∣∣ 1(1− bn−b

z−b

)n −M

∑k=0

ak

(bn−bz−b

)k∣∣∣∣∣∣< ε (dist(b,K))n

2. (56.1.7)

Therefore, for all z ∈ K ∣∣∣∣∣Qbn (z)−1

(z−b)n

M

∑k=0

ak

(bn−bz−b

)k∣∣∣∣∣ =∣∣∣∣∣∣ 1

(z−b)n(

1− bn−bz−b

)n −1

(z−b)n

M

∑k=0

ak

(bn−bz−b

)k∣∣∣∣∣∣ ≤

ε (dist(b,K))n

21

dist(b,K)n =ε

2