Divergent Sequences
Definition 15.1. Let $\{a_n\}$ be a sequence. We say that $\{a_n\}$ diverges to infinity (or minus infinity) and write \[ \lim_{n \to \infty} a_n = \infty \quad (\lim_{n \to \infty} a_n = -\infty) \] if for every real number $M$, there exists a positive integer $N$ such that if $n \geq N$, then $a_n > M$ ($a_n < M$).
Theorem 15.2
Theorem 15.2. Let $\{a_n\}$ and $\{b_n\}$ be sequences such that \[ \lim_{n \to \infty} a_n = \infty = \lim_{n \to \infty} b_n. \] Then \[ \lim_{n \to \infty} (a_n + b_n) = \infty. \]
Proof. Let $M \in \mathbb{R}$. Then $\exists N_1, N_2 \in \mathbb{P}$ such that $a_n > \frac{M}{2}, \forall n \geq N_1$ and $b_n > \frac{M}{2}, \forall n \geq N_2$. Let $N = \max \{N_1, N_2 \}$. Then $$a_n + b_n > \frac{M}{2} + \frac{M}{2} = M, \forall n \geq N.$$ Thus $\lim_{n \to \infty} (a_n + b_n) = \infty$. $\blacksquare$
Squeeze Theorem
Theorem 15.3 (Squeeze Theorem). Let $\{a_n\}$ and $\{b_n\}$ be sequences such that \[ a_n \leq b_n \quad \text{for every positive integer } n. \] If \[ \lim_{n \to \infty} b_n = -\infty, \] then \[ \lim_{n \to \infty} a_n = -\infty. \]
Proof. Let $M \in \mathbb{R}$. Then $\exists N \in \mathbb{P}$ such that $b_n < M, \forall n \geq N$. Then $a_n \leq b_n < M, \forall n \geq N$, therfore $\lim_{n \to \infty} a_n = - \infty$. $\blacksquare$