Problem 134
How many (a) maxima (b) upper bounds of a set of real numbers can have? Difficulty: 2. |
Problem 136
Determine the minimum, maximum, infimum, supremum of the following sets (if they have any)! (1) \(\displaystyle [1 , 2]\), (2) \(\displaystyle (1 , 2)\), (3) \(\displaystyle \{\frac1n: n \in \N^+ \}\), (4) \(\displaystyle \Q\), (5) \(\displaystyle \{\frac1n + {1 \over \sqrt{n}}: n \in \N^+ \}\), Difficulty: 2. |
Problem 137
Are the following sets bounded from above or from below? What is the maximum, minmimum, supremum and infimum? Which set is convex? \(\displaystyle \emptyset \qquad \{1,2,3,\dots\} \qquad \{1,-1/2,1/3,-1/4,1/5,\dots\} \qquad \Q \qquad \R \qquad [1,2) \qquad (2,3] \qquad [1,2)\cup(2,3] \) Difficulty: 2. |
Problem 138
Let \(\displaystyle H\) be a subset of the reals. Which properties of \(\displaystyle H\) are expressed by the following formulas? Difficulty: 2. |
Problem 129
Prove that \(\displaystyle \sqrt{2}\) is irrational. Difficulty: 3. |
Problem 131
Let \(\displaystyle a,b\in \Q\) and \(\displaystyle c,d\) be irrational. What can we say about the rationality of \(\displaystyle a+b\), \(\displaystyle a+c\), \(\displaystyle c+d\), \(\displaystyle ab\), \(\displaystyle ac\) and \(\displaystyle cd\)? Difficulty: 3. |
Problem 132
Prove that there is a rational and an irrational number in every open interval. Difficulty: 3. |
Problem 139
Let \(\displaystyle A\cap B\ne \emptyset\). What can we say about the connections among \(\displaystyle \sup A,\ \sup B\) and \(\displaystyle \sup (A\cup B) ,\ \sup(A\cap B)\) and \(\displaystyle \sup (A\setminus B)\)? Difficulty: 3. |
Problem 141
Which subsets \(\displaystyle H\subset\R\) satisfy that (a) \(\displaystyle \inf H<\sup H\); (b) \(\displaystyle \inf H=\sup H\); (c) \(\displaystyle \inf H>\sup H\)? Difficulty: 3. |
Problem 130
Prove that (1) \(\displaystyle \sqrt 3\) is irrational; (2) \(\displaystyle {\sqrt 2 \over \sqrt 3}\) is irrational; (3) \(\displaystyle {{{{\sqrt 2 + 1} \over 2} + 3} \over 4} + 5\) is irrational! Difficulty: 4. |
Problem 145
Let \(\displaystyle a_n=\sqrt{n+1}+(-1)^n\sqrt{n}\). \(\displaystyle \inf\{a_n|n\in \N\}=?\) Difficulty: 4. |
Problem 143
What are the suprema and infima of the following sets? a) \(\displaystyle \{{1\over n}|n\in\N\}\). b) \(\displaystyle \{{1\over n}|n\in\N\}\cup\{0\}\). c) \(\displaystyle \{{1\over n}|n\in\N\}\cup\{{-1\over n}|n\in\N\}\). d) \(\displaystyle \{{1\over n^n}|n\in\N\}\cup \{2,3\}\). e) \(\displaystyle \{{\cos n\over n^n}|n\in\N\}\cup [-6,-5] \cup (100,101)\). Difficulty: 5. |
Problem 144
Let \(\displaystyle H,K\) be non-empty subsets of the real line \(\displaystyle \R\). What is the logical connection between the following two statements? a) \(\displaystyle \sup H < \inf K\); b) \(\displaystyle \forall x\in H\) \(\displaystyle \exists y\in K\) \(\displaystyle x<y\). Difficulty: 5. |
Problem 146
Let \(\displaystyle A,B\) be subsets of the real line \(\displaystyle \R\) such that \(\displaystyle A\cup B=(0,1)\). Does it imply that \(\displaystyle \inf A=0\qquad\hbox{or}\qquad \inf B=0\qquad?\) Difficulty: 5. |
Problem 147
Prove that all convex subset of \(\displaystyle \R\) are intervals. Difficulty: 7. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |