Geometry
# Topology

Suppose $\{a_n\}$ is a sequence of real numbers such that $a_{n+1} = \frac{1}{2}a_n + 3$ for each natural number $n \geq 1.$ Which of these statements are true?

A. If $\displaystyle \lim_{n \to \infty}a_n = L,$ then $L = \frac{1}{2}L + 3.$

B. $\displaystyle \lim_{n \to \infty}a_n = 6.$

Suppose $\{a_n\}$ is a sequence of real numbers such that $a_1 \neq -3$ and $a_{n+1} = 2a_n + 3$ for each natural number $n \geq 1.$ Which of these statements are true?

A. If $\displaystyle \lim_{n \to \infty}a_n = L,$ then $L = 2L + 3.$

B. $\displaystyle \lim_{n \to \infty}a_n = -3.$

Suppose $\{a_n\}$ is a sequence of real numbers such that $a_{n+1} = \frac{1}{2}a_n + 3$ for each natural number $n \geq 1.$ Which of these statements are true?

A. $\{a_n\}$ is bounded.

B. $\{a_n\}$ is monotone.

C. $\{a_n\}$ converges.

**Definitions.**
A sequence $\{x_n\}$ of real number is:

*bounded* if there is a real number $M$ such that $-M < x_n < M$ for all $n.$

*monotone* if $x_n \leq x_{n+1}$ for all $n,$ or $x_n \geq x_{n+1}$ for all $n.$

*convergent* if there is a number $L$ such that for every $\epsilon > 0,$ there is a natural number $N$ satisfying

$n > N \Rightarrow |x_n - L| < \epsilon.$

Suppose $\{a_n\}$ is a sequence of real numbers such that $a_1 \neq -3$ and $a_{n+1} = 2a_n + 3$ for each natural number $n \geq 1.$ Which of these statements are true?

A. $\{a_n\}$ is bounded.

B. $\{a_n\}$ is monotone.

C. $\{a_n\}$ converges.

**Definitions.**
A sequence $\{x_n\}$ of real number is:

*bounded* if there is a real number $M$ such that $-M < x_n < M$ for all $n.$

*monotone* if $x_n \leq x_{n+1}$ for all $n,$ or $x_n \geq x_{n+1}$ for all $n.$

*convergent* if there is a number $L$ such that for every $\epsilon > 0,$ there is a natural number $N$ satisfying

$n > N \Rightarrow |x_n - L| < \epsilon.$

Which of these statements are true?

A. Every **bounded, monotone** sequence of real numbers converges

B. Every **bounded** sequence of real numbers converges

C. Every **monotone** sequence of real numbers converges

**Definitions.**
A sequence $\{x_n\}$ of real number is:

*bounded* if there is a real number $M$ such that $-M < x_n < M$ for all $n.$

*monotone* if $x_n \leq x_{n+1}$ for all $n,$ or $x_n \geq x_{n+1}$ for all $n.$

*convergent* if there is a number $L$ such that for every $\epsilon > 0,$ there is a natural number $N$ satisfying

$n > N \Rightarrow |x_n - L| < \epsilon.$