Problems 1–8 are required; problem 9 is optional.
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Decide which of the following sets are neighbourhoods of the point $x_{0} = - 1$:
a) $( - 2,0)$;
b) $\lbrack - 1,1\rbrack$;
c) $( - 3,1) \smallsetminus \left\{ 0 \right\}$;
d) $\mathbb{Q}$;
e) ${\mathbb{R}} \smallsetminus {\mathbb{Z}}$.
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Which of the following sets are neighbourhoods of $- \infty$?
a) $( - \infty, - 100)$;
b) $\mathbb{Z}$;
c) ${\mathbb{R}} \smallsetminus \lbrack - 1,1\rbrack$;
d) $( - \infty,0) \cap {\mathbb{Q}}$.
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Find $m \in {\mathbb{R}}$ for which the interval $V = \left( m - 1,m^{2} + 1 \right)$ is a neighbourhood of $2$.
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Find the set of accumulation points and the isolated points of:
a) $A = ( - 2,1\rbrack \cup \left\{ 3 \right\}$;
b) $B = \left\{ \frac{n}{n + 1}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}} \right\}$;
c) $C = {\mathbb{N}}$;
d) $D = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu x^{2} - 5x + 4 \leq 0 \right\} \cup \left\{ 6 \right\}$;
e) $E = (0, + \infty)$;
f) $F = \left\{ ( - 1)^{n} + \frac{1}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$.
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a) Show that if $V_{1}$ and $V_{2}$ are neighbourhoods of $+ \infty$, then $V_{1} \cap V_{2}$ is a neighbourhood of $+ \infty$.
b) Show that the sets $V_{n} = \left( - \frac{1}{n},\frac{1}{n} \right)$, $n \in {\mathbb{N}}^{*}$, are neighbourhoods of $0$, but the intersection of all these sets is not a neighbourhood of $0$.
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Let $A = \left\{ \frac{2n - 1}{n + 1}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}} \right\}$. Find $\min A$, $\sup A$ and $A'$.
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Show that $+ \infty$ is an accumulation point of the set $A \subset {\mathbb{R}}$ if and only if $A$ is not bounded above. Application: show that $+ \infty \in A'$ for $A = \left\{ n^{2} - 3n\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}} \right\}$.
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Decide whether each of the following statements is true or false; justify:
a) Every accumulation point of a set belongs to the set.
b) A finite set has no accumulation points.
c) Every point of the interval $(a,b)$ is an accumulation point of this interval.
d) The set $\mathbb{Z}$ is a neighbourhood of $0$.
e) If $V$ is a neighbourhood of $x_{0}$ and $W \subset {\mathbb{R}}$ is any set, then $V \cup W$ is a neighbourhood of $x_{0}$.
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(optional) Show that $x_{0} \in {\mathbb{R}}$ is an accumulation point of the set $A$ if and only if every neighbourhood of $x_{0}$ contains infinitely many elements of $A$.