Sequences Introduction
Definition
An infinite sequence of real numbers \(\{a_n\}_{n=1}^\infty\) is a function from the natural numbers into the reals. The infinite sequence is said to converge to \(L\) as \(n\) tends to infinity when
\[{}\forall \epsilon > 0, \exists N \in {\mathbb N} \text{ such that }{}\]
\[{}n \geq N \Rightarrow |a_n - L| < \epsilon.{}\]
In words, for all \(\epsilon\) greater than zero, there is some natural number \(N\) such that \(n\) being greater than or equal to \(N\) implies that the distance from \(a_n\) to \(L\) (measured by absolute value of the difference) is strictly smaller than \(\epsilon\). When \(\{a_n\}_{n=1}^\infty\) converges to \(L\) as \(n \rightarrow \infty\), we write
\[ \lim_{n \rightarrow \infty} a_n = L. \]
Lemma
If \(\{b_n\}_{n=1}^\infty\) is a convergent sequence, then the set
\[ \{ x \in {\mathbb R} \ : \ x = b_n \text{ for some } n \in {\mathbb N}\}\]
is bounded, i.e., there exists \(M\) such that \(|b_n| \leq M\) for all \(n \in N\).
Proof
If the limit is \(L\), fix \(\epsilon = 1\); we know that there is some \(N\) such that \(|b_n - L| < 1\) for all \(n \geq N\). The triangle inequality implies that \(|b_n| \leq |L| + 1\) for all such \(b \geq N\). However, the \(b_n\)'s not subject to this inequality are themselves finite, so there is a term \(|b_{n_0}|\) which is equal to the maximum \(\max\{|b_1|,\ldots,|b_{N-1}|\}\) when \(N > 1\) (if \(N =1\), we're in good shape because the inequality \(|b_n| \leq |L| + 1\) applies to all terms.) Thus if
\[ M := \max \{|b_1|,\ldots,|b_{N-1}|,|L|+1\},\]
it follows that \(|b_n| \leq M\) for all \(n \geq 1\).
Theorem (Algebraic Limit Laws)
If \(\lim_{n \rightarrow \infty} a_n = A\) and \(\lim_{n \rightarrow \infty} b_n = B\),
- \(\lim_{n \rightarrow \infty} (a_n + b_n) = A + B\).
- \(\lim_{n \rightarrow \infty} (a_n - b_n) = A - B\).
- \(\lim_{n \rightarrow \infty} a_n b_n = A B\).
- For any constant \(c\), \(\lim_{n \rightarrow \infty} c a_n = cA\).
- If \(B \neq 0\), \(\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \frac{A}{B}\). In particular, assuming that the limits of \(a_n\) and \(b_n\) exist, all the other listed limits must exist as well and have the stated values.
Proof
Definition
A sequence \(\{a_n\}_{n=1}^\infty\) of real numbers is called a Cauchy sequence when for every \(\epsilon > 0\), there exists some \(N \in {\mathbb N}\) such that \(|a_n - a_m| < \epsilon\) whenever both \(n,m \geq N\).
Theorem
A sequence of real numbers \(\{a_n\}_{n=1}^\infty\) is a Cauchy sequence if and only if it converges.
Proof
Postponed until after the Bolzano-Weierstrass Theorem. For the proof, see the upcoming section on Cauchy sequences.