Infi-1 5

Limit of the sequence #definition

\begin{matrix} a_{n} \to L \\ \forall ε > 0 \exists n_{ε} : \forall n > n_{ε} | a_{n} - L | < ε \end{matrix}

a_{n} \to L; b_{n} \to M

\begin{matrix} a_{n} \pm b_{n} \to L \pm M \\ a_{n} * b_{n} \to L * M \\ \frac{a_{n}}{b_{n}} \to \frac{L}{M} ∣ M \neq 0 \\ C * a_{n} \to C * L \\ \sqrt{a_{n}} \to \sqrt{L} ∣ a_{n} \geq 0 \\ | a_{n} | \to | L | \end{matrix}

Prove: \forall x, y \in R : | x + y | \leq | x | + | y |

\begin{matrix} | x + y | \leq | x | + | y | \leftrightarrow | x + y |^{2} \leq (| x | + | y |)^{2} \\ \leftrightarrow (x + y)^{2} \leq x^{2} + 2 | x | * | y | + y^{2} \\ \leftrightarrow 2 x * y \leq 2 | x | * | y | \leftrightarrow 2 x * y \leq | 2 x * y | \end{matrix}

Prove: | | x | - | y | | \leq | x - y |

Prove: a_{n} + b_{n} \to L + M

\begin{matrix} \exists n_{1} : \forall n > n_{1} : | a_{n} - L | < ε_{1} = \frac{ε}{2} \\ \exists n_{2} : \forall n > n_{2} : | b_{n} - L | < ε_{2} = \frac{ε}{2} \\ | a_{n} + b_{n} - (L + M) | = | a_{n} - L + b_{n} - M | \leq | a_{n} - L | + | b_{n} - M | \\ \exists n_{ε} = m a x (n_{1}, n_{2}) : \forall n > n_{ε} : | a_{n} + b_{n} - (L + M) | \leq | a_{n} - L | + | b_{n} - M | < ε_{1} + ε_{2} = \frac{ε}{2} + \frac{ε}{2} = ε \\ ⟹ \exists n_{ε} = m a x (n_{1}, n_{2}) : \forall n > n_{ε} : | a_{n} + b_{n} - (L + M) | < ε \\ ⟹ a_{n} + b_{n} \to L + M \end{matrix}

lim_{n \to + \infty} a_{n} = + \infty \leftrightarrow \exists M > 0 : \forall n > n_{M} : M < a_{n}

lim_{n \to \infty} a_{n} = - \infty \leftrightarrow lim_{n \to \infty} - a_{n} = + \infty

| a_{n} | \to 0 \leftrightarrow a_{n} \to 0

Prove: a_{n} > 0, \frac{1}{a_{n}} \to 0 ⟹ a_{n} \to + \infty

\begin{matrix} \exists n_{ε} : \forall n > n_{ε} : | \frac{1}{a_{n}} | < ε \leftrightarrow \frac{1}{| a_{n} |} < ε \leftrightarrow | a_{n} | > \frac{1}{ε} \\ ⟹ \exists M = \frac{1}{ε} : \forall n > n_{M} = n_{ε} : | a_{n} | > M ⟹ a_{n} \to \infty \end{matrix}