Page 125 - 'Blast_Into_Math

Blast into Math! Mathematical perspectives: all aour mase are melong to us

To help us get warmed up for playing mathematical baseball, let’s prove a lemma which we will need to
prove the main theorem of this chapter.

n
Lemma 3 (Base Lemma). For all n ∈ N , and for any base b , b >n .

Proof: Since the lemma is a statement for all n ∈ N, we can prove the lemma by induction. The base
case is n =1. By the definition of base, b ≥ 2. Is 2 >n when n =1? Well, yes, because in this case
n
1
2 =2 > 1. Since any base is at least as big as 2, we know that

1
1
b ≥ 2 > 1, forany base b.

We have proven the base case, and now we assume that b >n . Then, for the particular choice of base
n
b =2, when we multiply both sides of the inequality 2 >n by 2, we have by the exponent rules,
n
n
1
n
2 n+1 =2 ∗ 2 =2 ∗ 2 > 2 ∗ n = n + n ≥ n +1 because n ∈ N so n ≥ 1.

125