Chapter 4.4, Problem 6E

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.

1. Claim. Every bounded non-increasing sequence converges.
   "Proof." Let x be a bounded non-increasing sequence. Then $y_n=-x_n$ defines a bounded non-decreasing sequence. By the proof of Theorem 7.4.3, $\underset{n\to \infty }{\lim }{y}_n=L$ for some L. Thus, $\underset{n\to \infty }{\lim }{x}_n=-L.$
2. Claim. The sequence x, where $x_n=\frac{\ln n}{n},$ converges.

* Augustin Louis Cauchy (1789—1857) was a creative mathematician and pioneer in the efforts to bring rigor to the infinitesimal calculus. He was the first to define complex numbers as a pair of real numbers. Cauchy's name is associated with concepts and results in many fields of mathematics, includinggeometry, analysis, and mathematical physics.

"Proof." Because $\ln n\le n$ for all natural numbers n, $\frac{\ln n}{n}\le 1.$ Therefore, x is a bounded sequence. The derivative of $\frac{\ln n}{n}$ is $\frac{1-\ln n}{n^2}$ which is less than 0 for every natural number n greater than e. Therefore, except for the first two terms, x is a decreasing sequence. Because x is bounded and decreasing, x converges.