Zorich Mathematical Analysis Solutions

Zorich I, §1.2, Ex.5 — Show that the sequence a_n = (1 + 1/n)^n is increasing and bounded above by e.

A: No. A complete, error-free, unified solution manual does not exist publicly. The closest are Roitershtein’s notes (Ch. 1–6) and scattered GitHub repositories (Ch. 7–8 on integrals). zorich mathematical analysis solutions

have a vast archive of Zorich's problems already solved. Searching by the specific theorem name or problem statement usually yields a detailed breakdown. University Course Pages: Zorich I, §1

Solution: Let $x$ be a real number and $\epsilon > 0$. We need to show that there exists a rational number $q$ such that $|x - q| < \epsilon$. Since $x$ is a real number, there exists a sequence of rational numbers $q_n$ such that $q_n \to x$ as $n \to \infty$. Therefore, there exists $N$ such that $|x - q_N| < \epsilon$. Let $q = q_N$. Then $|x - q| < \epsilon$, which proves the result. The closest are Roitershtein’s notes (Ch