2.01- Define sequence of real numbers.
2.02- Define subsequence
2.03- Define increasing sequence.
2.04- Define decreasing sequence.
2.05- Define monotnone sequence.
2.06- Define bounded sequence.
2.07- Prove that $\{\frac{1}{n}\}$ is decreasing sequence.
2.08- Prove that $\{1+\frac{1}{n}\}$ is a decreasing sequence.
2.09- Prove that $\{\frac{n+1}{n+2}\}$ is increasing sequence.
2.10- Is the sequence $\{\frac{n+2}{n}\}$ is increasing or decreasing sequence?
2.11- Define convergence of the sequence.
2.12- By definition, prove that $\lim\limits_{n\to\infty}\frac{2n}{n+2}=2$.
2.13- By definition, prove that $\lim\limits_{n\to\infty}\frac{n^2-1}{2n^2+3}=\frac{1}{2}$.
2.14- By definition, prove that $\lim\limits_{n\to\infty}\frac{1}{3^n}=0$.
2.15- Prove that a convergent sequence of real number has one and only one limit.
2.16- Prove that the limit of the sequence is unique. (alternative statement of Q # 15)
2.17- Prove that if the sequence $\{s_n\}$ converges to $s$, where $s\neq 0$, then there exists a positive integer $n_1$ such that $|s_n|>\frac{1}{2}|s|$ for all $n>n_1$.
2.18- Prove that if the sequence $\{s_n\}$ converges to $s$, where $s\neq 0$, then there exists a positive integer $n_1$ such that $|s_n|>\frac{1}{5}|s|$ for all $n>n_1$.
2.19- Prove that if $\lim\limits_{n\to\infty}{s_n}=t$, then $\lim\limits_{n\to\infty}{|s_n|}=|t|$ but converse is not true in general.
2.20- Prove that every convergent sequence is bounded.
2.21- State and prove sandwich theorem.
2.22- State and prove squeeze theorem.
2.23- Suppose that $\{x_n\}$ and $\{y_n\}$ be two convergent sequences such that $\lim\limits_{n\to\infty} x_n=\lim\limits_{n\to\infty} y_n=c$. If $x_n<z_n<y_n$ for all $n>n_0$, then the sequence $\{z_n\}$ also converges to $c$.
2.24- Prove that for each irrational number $x$, there exists a sequence $\{r_n\}$ of distinct rational numbers such that $\lim\limits_{n\to\infty}r_n=x$.
2.25- Prove that a bounded increasing sequence converges to its supremum.
2.26- Prove that a bounded decreasing sequence converges to its infimum.
2.27- Let $\{t_n\}$ be a positive term sequence. Find the limit of the sequence if $4t_{n+1}=\frac{2}{5}-3t_n$ for all $n\geq 1$.
2.28- Let $\{u_n\}$ be a positive term sequence. Find the limit of the sequence if $u_{n+1}=\frac{1}{u_n}+\frac{1}{4}u_n$ for all $n\geq 1$. (consider other questions similar to this)
2.29- The Fibonacci numbers are: $F_1=F_2=1$ and for every $n\geq 3$, $F_n$ is defined by $F_n=F_{n-1}+F_{n-2}$. Find the $\lim\limits_{n\to\infty}\frac{F_n}{F_{n-1}}$.
2.30- Define Cauchy sequence.
2.31- By definition, prove that $\{\frac{1}{n}\}$ is a Cauchy sequence.
2.31- Prove that a Cauchy sequence of real numbers is bounded.
2.32- Prove that every Cauchy sequence of real number is bounded but converse is
not true.
2.33-Prove that every convergent sequence is bounded but converse is not true.
2.34- Prove that every Cauchy sequence of real numbers has a convergent subsequence.
2.35- Prove that a sequence of real number is convergent if and only if it is a Cauchy sequence.
2.36- Prove that $\{1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n} \}$ is divergent sequence.
2.37- Prove that $\{s_n \}$, where $s_n=\sum_{k=1}^n \frac{1}{n}$, is divergent.
2.38- Define limit inferior of the sequence.
2.39- Define limit superior of the sequence.
2.40- Prove that if $\{s_n\}$ is a convergent sequence, then $\lim\limits_{n\to\infty}s_n = \liminf\limits_{n\to\infty} s_n=\limsup\limits_{n\to\infty} s_n$.
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