32 CHAPTER 1. SOME PREREQUISITE TOPICS

Theorem 1.17.6 Suppose {an} is a sequence of points of [−∞,∞] . Let

λ = lim supn→∞

an.

Then if b > λ , it follows there exists N such that whenever n ≥ N, an ≤ b. If c < λ , thenan > c for infinitely many values of n. Let γ = liminfn→∞ an. Then if d < γ, it follows thereexists N such that whenever n ≥ N, an ≥ d. If e > γ, it follows an < e for infinitely manyvalues of n.

The proof of this theorem is left as an exercise for you. It follows directly from the defi-nition and it is the sort of thing you must do yourself. Here is one other simple proposition.

Proposition 1.17.7 Let limn→∞ an = a > 0. Then

lim supn→∞

anbn = a lim supn→∞

bn.

Proof: This follows from the definition. Let λ n = sup{akbk : k ≥ n} . For all n largeenough, an > a− ε where ε is small enough that a− ε > 0. Therefore,

λ n ≥ sup{bk : k ≥ n}(a− ε)

for all n large enough. Then

lim supn→∞

anbn = limn→∞

λ n ≡ lim supn→∞

anbn

≥ limn→∞

(sup{bk : k ≥ n}(a− ε))

= (a− ε) lim supn→∞

bn

Similar reasoning shows limsupn→∞ anbn ≤ (a+ ε) limsupn→∞ bn. Since ε > 0 is arbitrary,the conclusion follows. ■

1.18 Exercises

1. Prove by induction that ∑nk=1 k3 =

14

n4 +12

n3 +14

n2.

2. Prove by induction that whenever n≥ 2,∑nk=1

1√k>√

n.

3. Prove by induction that 1+∑ni=1 i(i!) = (n+1)!.

4. The binomial theorem states (x+ y)n = ∑nk=0(n

k

)xn−kyk where(

n+1k

)=

(nk

)+

(n

k−1

)if k ∈ [1,n] ,

(n0

)≡ 1≡

(nn

)Prove the binomial theorem by induction. Next show that(

nk

)=

n!(n− k)!k!

, 0!≡ 1

▶