Problem 259
Given \(\displaystyle a_1,\ldots, a_p\in \R\), find a sequence with exactly these limit points. Difficulty: 1. |
Problem 262
Calculate the limit points of the sets \(\displaystyle B(0,1)\), \(\displaystyle \dot{B}(0,1)\), \(\displaystyle \N\), \(\displaystyle \Q\) and \(\displaystyle \{ 1/n:n\in\N \}\)! Difficulty: 2. |
Problem 267
What are the limit points, limsup and liminf of the following sequences? \(\displaystyle \root{n}\of{n}; \qquad (-1)^n+\frac1n; \qquad \big\{\sqrt{n}\big\} \) Difficulty: 2. |
Problem 268
What is the limsup and liminf of the following sequence? \(\displaystyle a_n={n^k \over 2^n}.\) Difficulty: 2. |
Problem 258
Find a non convergent sequence with exactly one limit point. Difficulty: 3. Solution is available for this problem. |
Problem 272
Prove that if \(\displaystyle a_n\to a>0\) and \(\displaystyle (b_n)\) is an arbitrary sequence, then \(\displaystyle \Ulim (a_n\cdot b_n) = a \cdot \Ulim b_n \quad\text{and}\quad \Olim (a_n\cdot b_n) = a \cdot \Olim b_n. \) Difficulty: 3. |
Problem 269
Using the definition of \(\displaystyle \limsup\) and \(\displaystyle \liminf\) prove that \(\displaystyle \liminf a_n \le \limsup a_n\). Difficulty: 4. |
Problem 271
Prove that if \(\displaystyle (a_n)\) is convergent and \(\displaystyle (b_n)\) is an arbitrary sequence, then \(\displaystyle \Olim (a_n+b_n) = \lim a_n + \Olim b_n. \) Difficulty: 4. |
Problem 274
Prove that if the sequence \(\displaystyle (a_n)\) is bounded with \(\displaystyle \liminf a_n>0\) and \(\displaystyle b_n\to0\), then \(\displaystyle a_n^{b_n}\to1\). Difficulty: 4. |
Problem 263
Prove that the set of limit points of a sequence (or a set) is closed. Difficulty: 5. |
Problem 273
Prove that if (i) \(\displaystyle a_n\to a\ge1\) and \(\displaystyle (b_n)\) is bounded, then \(\displaystyle \Olim a_n^{b_n} = a^{\Olim b_n} \quad\text{and}\quad \Ulim a_n^{b_n} = a^{\Ulim b_n}. \) (ii) \(\displaystyle a_n\to a\le 1\) and \(\displaystyle (b_n)\) is bounded, then \(\displaystyle \Olim a_n^{b_n} = a^{\Ulim b_n} \quad\text{and}\quad \Ulim a_n^{b_n} = a^{\Olim b_n}. \) Difficulty: 5. |
Problem 275
Bizonyítsuk be, hogy tetszőleges \(\displaystyle a_1,a_2,\ldots\) valós számsorozatra \(\displaystyle \liminf\frac{a_1+a_2+\ldots+a_n}{n} \ge \liminf a_n \quad\text{és}\quad \limsup\frac{a_1+a_2+\ldots+a_n}{n} \le \limsup a_n. \) Difficulty: 5. |
Problem 276
Igazoljuk, hogy ha \(\displaystyle a_n\to a\), akkor \(\displaystyle \inf \big\{\sup\{a_n,a_{n+1},a_{n+2},\ldots\}: n\in\N \big\} = a. \) Difficulty: 5. |
Problem 264
Find a sequence such that that the set of limit points of it is \(\displaystyle [0,1]\). Difficulty: 6. Solution is available for this problem. |
Problem 265
Prove that a limit point of the set of limit points of a set is a limit point of the original set. Difficulty: 6. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |