19.4. THE MÜNTZ THEOREM 533

which shows

d =1√

2m+1

n

∏k=1

|m− pk|m+ pk +1

.

and this proves the lemma.The following lemma relates an infinite sum to a product. First consider the graph of

ln(1− x) for x ∈[0, 1

2

]. Here is a rough sketch with two lines, y =−x which lies above the

graph of ln(1− x) and y =−2x which lies below.

12

Lemma 19.4.3 Let an ̸= 1,an > 0, and limn→∞ an = 0. Then

∞

∏k=1

(1−an)≡ limn→∞

n

∏k=1

(1−an) = 0

if and only if∞

∑n=1

an =+∞.

Proof:Without loss of generality, you can assume an < 1/2 because the two condi-tions are determined by the values of an for n large. By the above sketch the following isobtained.

lnn

∏k=1

(1−ak) =n

∑k=1

ln(1−ak) ∈

[−2

n

∑k=1

ak,−n

∑k=1

ak

].

Therefore,

e−2∑nk=1 ak ≤

n

∏k=1

(1−ak)≤ e−∑nk=1 ak

The conclusion follows.The following is Müntz’s first theorem.

Theorem 19.4.4 Let {pn} be a sequence of real numbers larger than −1/2 such thatlimn→∞ pn = ∞. Let S denote the set of finite linear combinations of the functions

{xp1 ,xp2 , · · ·} .