Determine whether countable unions of countable sets are countable or not. In the affirmative give a proof, in the negative a counterexample.
Give an example of a bijection between $\mathbb{R}$ and $\mathbb{R}\setminus\mathbb{Z}$.
Is the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$ countable? Justify your answer.
Prove that the set of all open intervals in $\mathbb{R}$ has the same cardinality as $\mathbb{R}$.
Assuming that $\lim_{n\to\infty} x_n=x$ and that $\lim_{n\to\infty} y_n=y$, prove that $$ \lim_{n\to\infty} \frac{x_1y_n+\dots+x_n y_1}{n}=x y. $$
Let $x_1=\sqrt{2}$ and define $(x_n)_{n\in \mathbb{N}}$ recursively by $$ x_{n+1}=\bigl(2+x_n^{1/2}\bigr)^{1/2},\: n\geq 1. $$ Prove that the sequence converges.
Given real sequences $(x_n)_{n\in \mathbb{N}}$ and $(y_n)_{n\in \mathbb{N}}$, prove or disprove: $$ \liminf _{n\to\infty}(x_n+y_n)=\liminf _{n\to\infty}x_n+\liminf _{n\to\infty}y_n\, . $$
Investigate the behavior (convergence and divergence) of
$\sum_{n=1}^\infty a_n$ for
(a) $a_n=\sqrt{n+1}-\sqrt{n}$
(b) $a_n=\frac{\sqrt{n+1}-\sqrt{n}}{n}$
(c) $a_n=( n^{1/n}-1)^n$
(d) $a_n=\frac{1}{1+ z^n}$, where $z$ is a complex number.
Let $a_n\ge 0$ for $n\in \mathbb{N}$ and prove that $\sum_{n=1}^\infty \frac{\sqrt{a_n}}{n}$ converges if $\sum_{n=1}^\infty a_n$ does.
Let $(a_n)_{n\in \mathbb{N}}$ be a monotonically decreasing sequence of positive numbers and prove that $$ \sum _{n\in \mathbb{N}} a_n\text{ converges if and only if } \sum _{j\in\mathbb{N}}2^ja_{2^j}\text{ does.} $$