The Forty

1. State the definition of a convergent sequence in $ℝ$ and its limit.

2. State the definition of a Cauchy-sequence in $ℝ$.

3. Let $\overline{x}={\{x_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of real numbers. State the definition of the limitpoint set of $\overline{x}$.

4. State the definition of a closed set in $ℝ$.

5. State the Sequential Definition for continuity of a function $f:ℝ\to ℝ$ at $x\in ℝ$.

6. State the $(\epsilon ,\delta )$-Definition for continuity of a function $f:ℝ\to ℝ$ at $x\in ℝ$.

7. What does it mean that a function $f:ℝ\to ℝ$ is invertible?

8. What does “$x$ is an upper bound of a set $S\in ℝ$” mean ? State the Smallest Upper Bound Principle.

9. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of real numbers. What does the boundedness of this sequence mean?

10. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of real numbers. What does the limsup of this sequence mean?

11. Let ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{b_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences of positive numbers that tend to infinity. What does it mean that the sequence ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ beats the sequence ${\{b_n\}}_{n=1}^{\mathrm{\infty }}$?

12. State the Bolzano-Weierstrass Theorem.

13. State the Intermediate Value Theorem.

14. What does it mean that a sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ tend to infinity?

15. Give an example of a bounded sequence ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ so that: ${lim\; sup}_{n\to \mathrm{\infty }}{a}_n={lim\; inf}_{n\to \mathrm{\infty }}+2$.

16. Give an example of two bounded sequences ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{b_n\}}_{n=1}^{\mathrm{\infty }}$ such that ${lim\; sup}_{n\to \mathrm{\infty }}(a_n+b_n)\ne {lim\; sup}_{n\to \mathrm{\infty }}{a}_n+{lim\; sup}_{n\to \mathrm{\infty }}{b}_n$.

17. Give an example of an unbounded closed set.

18. Give an example of a function $f:ℝ\to ℝ$ that is continuous at $x$ if $x$ is an irrational number, but it is not continuous if $x$ is a rational number.

19. Give an example of a bounded sequence ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ having exactly $2$ limitpoints.

20. Give examples of:

    1. a bounded sequence that is not convergent.

    2. A sequence that is unbounded but it does not converges to infinity.

21. Give an example of a bounded set that has no maximum, but has a minimum.

22. Give an example of a sequence of irrational numbers that converge to a rational number.

23. Give an example of two positive sequences ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ and ${\{b_n\}}_{n=1}^{\mathrm{\infty }}$ tending to infinity such that

    1. ${\{a_n-b_n\}}_{n=1}^{\mathrm{\infty }}$ is bounded but not convergent.

    2. ${\{a_n-b_n\}}_{n=1}^{\mathrm{\infty }}$ is unbounded.

24. Give an example of a function $f:ℝ\to ℝ$ that is not everywhere continuous, but it is invertible.

25. Prove that any convergent sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ is bounded.

26. Prove that the sum of two Cauchy-sequences is still a Cauchy-sequence.

27. Prove that a sequence of positive real numbers ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ tends to $0$ if and only if the sequence ${\{\frac{1}{x_n}\}}_{n=1}^{\mathrm{\infty }}$ tends to infinity.

28. Prove that the set of rational numbers $\mathbb{Q}$ is not closed.

29. Prove that if $F_1$, $F_2$, $\mathrm{\dots }$ are all closed sets then their intersection ${\cap }_{n=1}^{\mathrm{\infty }}{F}_n$ is closed as well.

30. Prove that the set of zeros of a continuous function $f:ℝ\to ℝ$ is a closed set.

31. Prove that if $f,g:ℝ\to ℝ$ are continuous at $x\in ℝ$, then $f+g$ is continuous at $x\in ℝ$ as well.

32. Prove that a continuous function $f:[0,1]\to ℝ$ is bounded.

33. Prove that the sequence ${\{n!\}}_{n=1}^{\mathrm{\infty }}$ beats the sequence ${\{2^n\}}_{n=1}^{\mathrm{\infty }}$.

34. Prove that the interval $[0,1]$ is closed.

35. Prove that the interval $(0,1]$ is not closed.

36. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ be a bounded sequence and ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence tending to $0$. Prove that $x_n{y}_n\to 0$.

37. Let $\overline{x}={\{x_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of real numbers. Prove that the limitpoint set of $\overline{x}$ is a closed set.

38. Show that there exists a real number $0\le x\le 2$ so that $x^7+8x^2-10=0$.

39. Consider the sequences ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ and ${\{b_n\}}_{n=1}^{\mathrm{\infty }}$ defined by $a_n=\frac{n^2+1}{n}$ and $b_n=\frac{1}{a_n}$.

    1. State whether the sequence ${\{a_n\}}^{\mathrm{\infty }}$ is 1. bounded 2. convergent 3. increasing. (Proof is not required)

    2. Using statements of the lectures show that $b_n\to 0$.

    3. Let ${\{c_n\}}_{n=1}^{\mathrm{\infty }}$ be defined as $c_n=\frac{a_n+1}{a_n}$. Does this sequence converge? Explain briefly. If it converges, compute the limit.

40. Calculate ${lim}_{n\to \mathrm{\infty }}\frac{2n^3+n^2\sin (n)+6}{3n^3+n^2\cos (n)+10n+9}$. (Proof is not required)