82 CHAPTER 3. SEQUENCES AND COMPACTNESS

Definition 3.3.1 A sequence {an}∞

n=1 converges to a, written

limn→∞

an = a or an → a

if and only if for every ε > 0 there exists nε such that whenever n ≥ nε ,

|an −a|< ε.

Here a and an are assumed to be real numbers but the same definition holds more generally.

In words the definition says that given any measure of closeness ε, the terms of thesequence are eventually this close to a. Here, the word “eventually” refers to n beingsufficiently large. The above definition is always the definition of what is meant by thelimit of a sequence. However, in practice we usually say that something happens for nsufficiently large rather than trying to specify a particular size for how large n must be.First is a situation where the limit always exists. Nor do we determine limits by doingexperiments with calculators.

Proposition 3.3.2 Let {an}∞

n=1 be an increasing sequence meaning an ≤ an+1 for all nand suppose a ≡ sup{an : n ≥ 1} < ∞. Then limn→∞ an = a. A similar result holds if thesequence is decreasing and bounded below if a ≡ inf{an : n ≥ 1}.

Proof: For each ε > 0, there exists an ∈ [a− ε,a] since otherwise a is not equal towhat it is defined to be. Since {an} is increasing, it follows that an ∈ [a− ε,a] for all nlarge enough. Hence limn→∞ an = a. The situation where the sequence is decreasing andbounded below is exactly similar.

Next is the important theorem that the limit, if it exists, is unique.

Theorem 3.3.3 If limn→∞ an = a and limn→∞ an = â then â = a.

Proof: Suppose â ̸= a. Then let 0 < ε < |â−a|/2 in the definition of the limit. Itfollows that there exists nε such that if n ≥ nε , then |an −a| < ε and |an − â| < ε. Justlet nε be the larger of two numbers, one which works for a and one which works for â.Therefore, for such n,

|â−a| ≤ |â−an|+ |an −a|< ε + ε < |â−a|/2+ |â−a|/2 = |â−a| ,

a contradiction.

Example 3.3.4 Let an =1

n2+1 .

The intermediate value theorem from calculus is due to him. This was of interest because of attempts to provethe fundamental theorem of algebra. These days, the intermediate value theorem is considered obvious and isnot discussed well in calculus texts, but Bolzano knew better and gave a proof which identified exactly what wasneeded instead of relying on vague intuition and geometric speculation.

Like many of the other mathematicians, he was concerned with the notion of infinitesimals which had beenpopularized by Leibniz. Some tried to strengthen this idea and others sought to get rid of it. They realizedthat something needed to be done about this fuzzy idea. Bolzano was one who contributed to removing it fromcalculus. He also proved the extreme value theorem in 1830’s and gave the first formal εδ description of continuityand limits. This notion of infinitesimals did not completely vanish. These days, it is called non standard analysis.It can be made mathematically respectable but not in this book.