Real analysis/Section 2 Excercises

Exercises

1. Use the definition of convergence to prove that the following sequences converge/diverge and find their limits:
   1. $\left(\frac{x}{n}\right)$; $x\in ℝ$
   2. $\left(\frac{n-1}{n}\right)$
   3. $\left(\frac{2n-7}{n^2+1}\right)$
   4. $\left(x^n\right)$; $x>0$
   5. $\left(\frac{1}{n^x}\right)$; $x\in ℝ$
   6. $\left(\frac{\sqrt{2n-1}+2}{\sqrt{n+3}}\right)$
   7. The recursive sequence defined by the following:

      $x_0=2$

      $x_n=\sqrt{x_{n-1}}$

   8. The sequence of Cesaro Means for (1, 1, -1, 1, 1, -1...)
   9. $\underset{n\to }{\lim }sup(x_n)$ where $(x_n)=(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{3}{4},...)$
   10. Given any real number c, find a recursively defined sequence that converges to $\sqrt{c}$. (Hint: Use Newton's Method to find approximate the zeros of the function x^2-a. If you don't know about derivatives yet, just skip this and come back to it when you get there)
2. Prove the following statements about limits:
   1. If every convergent subsequence of $(x_n)$ converges to x, then $(x_n)\to x$
   2. $\underset{n\to \mathrm{\infty }}{\lim }(x_n)=0$ if and only if $\underset{n\to \mathrm{\infty }}{\lim }\left(\frac{1}{x_n}\right)=\mathrm{\infty }$
   3. If $(x_n)\to 0$ and $(y_n)$ is bounded, then $\underset{n\to \mathrm{\infty }}{\lim }(x_n{y}_n)=0$.
   4. Show that each of the equivalent formulations of completeness(Monotone Convergence Theorem, Nested Interval Property, Balzano-Weierstrass, Cauchy Convergence) are, in fact, equivalent. That is, show that we can arbitrarily take one of them as an axiom and directly deduce all the others. If you don't have that kind of time on your hands, just show that the original completeness axiom follows from Cauchy Convergence (this takes care of most of the cases anyway, if you go back and look at the proofs).
3. Prove that all constant sequences are Cauchy.
4. Prove that all null sequences are Cauchy.
5. Prove that the sum of Cauchy sequences is Cauchy.
6. Prove the product of Cauchy sequences is Cauchy.

More exercises to come...