56.1. RUNGE’S THEOREM 1771

Lemma 56.1.8 Let R be a rational function which has a pole only at a ∈ V, a componentof C \K where K is a compact set. Suppose b ∈ V. Then for ε > 0 given, there exists arational function Q, having a pole only at b such that

||R−Q||K,∞ < ε. (56.1.4)

If it happens that V is unbounded, then there exists a polynomial, P such that

||R−P||K,∞ < ε. (56.1.5)

Proof: Say that b ∈ V satisfies P if for all ε > 0 there exists a rational function, Qb,having a pole only at b such that

||R−Qb||K,∞ < ε

Now define a set,S≡ {b ∈V : b satisfies P } .

Observe that S ̸= /0 because a ∈ S.I claim S is open. Suppose b1 ∈ S. Then there exists a δ > 0 such that∣∣∣∣b1−b

z−b

∣∣∣∣< 12

(56.1.6)

for all z ∈ K whenever b ∈ B(b1,δ ) . In fact, it suffices to take |b−b1| < dist(b1,K)/4because then ∣∣∣∣b1−b

z−b

∣∣∣∣ <

∣∣∣∣dist(b1,K)/4z−b

∣∣∣∣≤ dist(b1,K)/4|z−b1|− |b1−b|

≤ dist(b1,K)/4dist(b1,K)−dist(b1,K)/4

≤ 13<

12.

Since b1 satisfies P, there exists a rational function Qb1 with the desired properties. Itis shown next that you can approximate Qb1 with Qb thus yielding an approximation to Rby the use of the triangle inequality,∣∣∣∣R−Qb1

∣∣∣∣K,∞

+∣∣∣∣Qb1 −Qb

∣∣∣∣K,∞≥ ||R−Qb||K,∞ .

Since Qb1 has poles only at b1, it follows it is a sum of functions of the form αn(z−b1)

n .

Therefore, it suffices to consider the terms of Qb1 or that Qb1 is of the special form

Qb1 (z) =1

(z−b1)n .

However,1

(z−b1)n =

1

(z−b)n(

1− b1−bz−b

)n