1880 CHAPTER 59. BASIC PROBABILITY

By the Borel Cantelli lemma, Lemma 59.1.2, there is a set of measure 0, N such that forω /∈ N, ω is in only finitely many of the sets,[

maxk≥m≥pn

∣∣∣∣∣ k

∑j=m

X j

∣∣∣∣∣≥ 2−n

]

and so for ω /∈ N, it follows that for large enough n,[max

k≥m≥pn

∣∣∣∣∣ k

∑j=m

X j (ω)

∣∣∣∣∣< 2−n

]

However, this says the partial sums{

∑kj=1 X j (ω)

}∞

k=1are a Cauchy sequence. Therefore,

they converge.With this amazing result, there is a simple proof of the strong law of large numbers. In

the following lemma, sk and a j could have values in any normed linear space.

Lemma 59.7.4 Suppose sk→ s. Then

limn→∞

1n

n

∑k=1

sk = s.

Also if∞

∑j=1

a j

j

converges, then

limn→∞

1n

n

∑j=1

a j = 0.

Proof: Consider the first part. Since sk → s, it follows there is some constant, C suchthat |sk|<C for all k and |s|<C also. Choose K so large that if k ≥ K, then for n > K,

|s− sk|< ε/2.∣∣∣∣∣s− 1n

n

∑k=1

sk

∣∣∣∣∣≤ 1n

n

∑k=1|sk− s|

=1n

K

∑k=1|sk− s|+ 1

n

n

∑k=K|sk− s|

≤ 2CKn

+ε

2n−K

n<

2CKn

+ε

2Therefore, whenever n is large enough,∣∣∣∣∣s− 1

n

n

∑k=1

sk

∣∣∣∣∣< ε.