Sequences of Real Numbers

We call an extended real number a cluster point of a sequence $\left\{ a_{n}\right\}$ if a subsequence converges to this extended real number. Show that $\liminf \left\{ a_{n}\right\}$ is the smallest cluster point of $\left\{ a_{n}\right\}$ and $\limsup \left\{ a_{n}\right\}$ is the largest cluster point of $\left\{ a_{n}\right\}$.

Prove Proposition 19.

Proposition 19: Let $\left\{ a_{n}\right\}$ and $\left\{ b_{n}\right\}$ be sequences of real numbers.

1. $\limsup \left\{ a_{n}\right\} = \ell \in \mathbb {R}$ if and only if for each $\epsilon > 0$, there are infinitely many indices $n$ for which $a_{n} > \ell - \epsilon$ and only finitely many indices $n$ for which $a_{n} > \ell + \epsilon$.

2. $\limsup \left\{ a_{n}\right\} = \infty$ if and only if $\left\{ a_{n}\right\}$ is not bounded above.

3. $\limsup \left\{ a_{n}\right\} = -\liminf \left\{ -a_{n}\right\}$.

4. A sequence of real numbers $\left\{ a_{n}\right\}$ converges to an extended real number $a$ if and only if $\liminf \left\{ a_{n}\right\} = \limsup \left\{ a_{n}\right\} = a$.

5. If $a_{n} \leq b_{n}$ for all $n$, then $\limsup \left\{ a_{n}\right\} \leq \limsup \left\{ b_{n}\right\}$.

Show that a sequence $\left\{ a_{n}\right\}$ is convergent to an extended real number if and only if there is exactly one extended real number that is a cluster point of the sequence.

Show that $\liminf a_{n} \leq \limsup a_{n}$.

Prove that if, for all $n, a_{n} \geq 0$ and $b_{n} \geq 0$, then

provided the product on the right is not of the form $0 \cdot \infty$.

Show that every real sequence has a monotone subsequence. Use this to provide another proof of the Bolzano-Weierstrass Theorem.

Let $p$ be a natural number greater than 1 , and $x$ a real number, $0 \leq x \leq 1$. Show that there is a sequence $\left\{ a_{n}\right\}$ of integers with $0 \leq a_{n} < p$ for each $n$ such that

and that this sequence is unique except when $x$ is of the form $q / p^{n}, 0 < q < p^{n}$, in which case there are exactly two such sequences. Show that, conversely, if $\left\{ a_{n}\right\}$ is any sequence of integers with $0 \leq a_{n} < p$, the series

converges to a real number $x$ with $0 \leq x \leq 1$. If $p = 10$, this sequence is called the decimal expansion of $x$. For $p = 2$ it is called the binary expansion; and for $p = 3$, the ternary expansion.

Prove Proposition 20.

Proposition 20: Let $\left\{ a_{n}\right\}$ be a sequence of real numbers.

1. The series $\sum_{k=1}^{\infty } a_{k}$ is summable if and only if for each $\epsilon > 0$, there is an index $N$ for which $\left|\sum_{k=n}^{n+m} a_{k}\right| < \epsilon$ for $n \geq N$ and any natural number $m$.

2. If the series $\sum_{k=1}^{\infty } \left|a_{k}\right|$ is summable, then $\sum_{k=1}^{\infty } a_{k}$ also is summable.

3. If each term $a_{k}$ is non-negative, then the series $\sum_{k=1}^{\infty } a_{k}$ is summable if and only if the sequence of partial sums is bounded.

Show that the assertion of the Bolzano-Weierstrass Theorem is equivalent to the Completeness Axiom for the real numbers. Show that the assertion of the Monotone Convergence Theorem is equivalent to the Completeness Axiom for the real numbers.