Table of Contents

An Example

Monotonicity

Monotone Sequence:

  • (\(a_n\)) is increasing if \(a_n \leq a_{n+1} \forall n\)
  • (\(a_n\)) is decreasing if \(a_n \geq a_{n+1} \forall n\)

Theorem: Any bounded monotone sequence converges.

Proof: \((a_n)\) bounded increasing. Let \(A=\{a_n | n \in \mathbb{N}\}\). We note that \(A\) is bounded, so there exists \(Z := \sup A\).

Claim: \(z = \lim_{n \to \infty} a_n\).

Idea: We must find an \(\epsilon\) such that \((z - \epsilon, z + \epsilon)\) contains \(z\).

  1. There is some \(a_n\) such that \(a_n \in (z - \epsilon, z + \epsilon)\)
    • This is true because if no such element exists, then \(z - \epsilon\) must be \(\sup A\). This contradicts the claim that \(z = \sup A\).
    • A consequence is that we know there exists at least one \(a_n\) within \((z - \epsilon, z)\).
  2. Then we will show \(\forall \epsilon > 0, \exists N > 0 \textrm{ s.t. } n > N \implies \vert a_n - a \vert < \epsilon\).

Proof: \(\forall \epsilon > 0\), we claim that \(\exists n > 0 \textrm{ s.t. } a_N \in (z - \epsilon, z + \epsilon)\). Then, \(\forall n > N\), we have \(z - \epsilon < a_N \leq a_n \leq z\). This implies that \(\vert a_n - z \vert < \epsilon\ \forall n > N\).

We note that sequences do not have to be monotone to converge.

\[(1, \frac{-1}{2}, \frac{1}{3}, \frac{-1}{4}, ...)\]

converges to \(0\).

Some Limit Theorems

If \(\lim_{n \to \infty} a_n = a, \lim{n \to \infty} b_n = b\) thm.

  1. \(r \neq 0, \lim_{n \to \infty} ra_n = ra\)
  2. \(\lim_{n \to \infty}a_n + b_n = a+b\)
  3. \(\lim_{n \to \infty} a_nb_n = ab\)
  4. If \(b_n \neq 0, \forall n, b\neq 0, then \lim_{n \to \infty} \frac{a_n}{b_n} = \frac{a}{b}\).

Example

Let us take \(a_n = \frac{4n^2 - 7n}{n^2 + 1}\). We can divide top and bottom by \(n^2\). Then, the top and bottom are both sequences. We take of their limits.

\[\lim_{n \to \infty} a_n = \frac{\lim (4 - \frac{7}{n})}{\lim (1 - \frac{1}{n^2})} = 4\]

Example 2

Let us take:

\[a_1 = 1\] \[a_2 = 1 + \frac{1}{1 + \frac{1}{1}}\] \[a_{n+1} = 1 + \frac{1}{1 + \frac{1}{a_n}}\] \[a_1 = 1\] \[a_2 = \frac{3}{2}\] \[a_3 = \frac{8}{5}\] \[a_4 = \frac{21}{13}\]

Proof that \((a_n)\) is bounded and increasing:

  • Bounded: \(1 < a_n < 2\).
  • Increasing: \(a_n \leq 1 + \frac{1}{1 + \frac{1}{a_n}} = \frac{2a_n+1}{a_n+1}\)