Last updates: 7 December 2009

Define the following and give an example for each:
- (a) sequence,
- (b) converges (for a sequence),
- (c) diverges (for a sequence),
- (d) limit (of a sequence),
- (e) sup (of a sequence),
- (f) inf (of a sequence),
- (g) lim sup (of a sequence),
- (h) lim inf (of a sequence),
- (i) bounded (for a sequence),
- (j) increasing (for a sequence),
- (k) decreasing (for a sequence),
- (l) monotone (for a sequence),
- (m) Cauchy sequence,
- (m) contractive sequence,
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Prove that if
$\left({a}_{n}\right)$
converges then
$\mathrm{lim}}_{n\to \infty}{a}_{n$
is unique.
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Prove that if
$\left({a}_{n}\right)$
converges then
$\left({a}_{n}\right)$
is bounded.
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Prove that if
${\mathrm{lim}}_{n\to \infty}{a}_{n}=a$
and
${\mathrm{lim}}_{n\to \infty}{b}_{n}=b$
then
${\mathrm{lim}}_{n\to \infty}{a}_{n}+{b}_{n}=a+b$. | |

Prove that if
${\mathrm{lim}}_{n\to \infty}{a}_{n}=a$
and
${\mathrm{lim}}_{n\to \infty}{b}_{n}=b$
then
${\mathrm{lim}}_{n\to \infty}{a}_{n}{b}_{n}=ab$. | |

Prove that if
${\mathrm{lim}}_{n\to \infty}{a}_{n}=a$
and
${\mathrm{lim}}_{n\to \infty}{b}_{n}=b$
and
${b}_{n}\ne 0$
for all
$n\in {\mathbb{Z}}_{>0}$
then
${\mathrm{lim}}_{n\to \infty}}\frac{{a}_{n}}{{b}_{n}}=\frac{a}{b$. | |

Prove that if
${\mathrm{lim}}_{n\to \infty}{a}_{n}=\ell$
and
${\mathrm{lim}}_{n\to \infty}{c}_{n}=\ell$
and
${a}_{n}\le {b}_{n}\le {c}_{n}$
for all
$n\in {\mathbb{Z}}_{>0}$
then
${\mathrm{lim}}_{n\to \infty}}{b}_{n}=\ell $. | |

Prove that if
$\left({a}_{n}\right)$
is increasing and bounded above
then
$\left({a}_{n}\right)$
converges.
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Prove that if
$\left({a}_{n}\right)$
is increasing and not bounded above
then
$\left({a}_{n}\right)$
diverges.
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Prove that if
$\left({a}_{n}\right)$
is decreasing and bounded below
then
$\left({a}_{n}\right)$
converges.
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Prove that if
$\left({a}_{n}\right)$
is decreasing and not bounded below
then
$\left({a}_{n}\right)$
diverges.
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Prove that every sequence
$\left({a}_{n}\right)$
of real numbers has a monotonic subsequence.
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(Bolzano-Weirstrass) Prove that every sequence
$\left({a}_{n}\right)$
of real or complex numbers has a convergent subsequence.
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Prove that every Cauchy sequence
$\left({a}_{n}\right)$
of real or complex numbers converges.
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Prove that every convergent sequence
$\left({a}_{n}\right)$
is a Cauchy sequence.
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Graph and determine the sup, inf, lim sup, lim inf and convergence of the following sequences:
- (a) ${a}_{n}={\left(-1\right)}^{n}$,
- (b) $a}_{n}=\frac{1}{n$,
- (c) $a}_{n}=\frac{{\left(n\text{!}\right)}^{2}{5}^{n}}{\left(2n\right)\text{!}$,
- (d) ${a}_{1}=3$, ${a}_{n}=\frac{1}{2}{\displaystyle ({a}_{n-1}+\frac{5}{{a}_{n-1}})}$,
- (e) ${a}_{n}={(1+\frac{1}{n})}^{n}$,
- (f) ${a}_{n}={e}^{in\pi /7}$,
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Does the sequence given by
$\frac{n}{2n+1}$
converge? If so, what is the limit?
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Does the sequence given by
$\sqrt{n}$
converge? If so, what is the limit?
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Does the sequence given by
$\frac{1}{\sqrt{n}}$
converge? If so, what is the limit?
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Does the sequence given by
$\sqrt{n+1}-\sqrt{n}$
converge? If so, what is the limit?
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Does the sequence given by
$\sqrt{n}(\sqrt{n+1}-\sqrt{n})$
converge? If so, what is the limit?
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Does the sequence given by
$\frac{n}{{n}^{2}+1}$
converge? If so, what is the limit?
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Does the sequence given by
$\frac{2n}{n+1}$
converge? If so, what is the limit?
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Does the sequence given by
$\frac{3n+1}{2n+5}$
converge? If so, what is the limit?
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Does the sequence given by
$\frac{{n}^{2}-1}{2{n}^{2}+3}$
converge? If so, what is the limit?
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Show that the sequence
${a}_{n}={\displaystyle {(1+\frac{1}{n})}^{n}}$
is increasing and bounded above by 3.
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Let
$a\in \mathbb{R}$
with
$\left|a\right|<1$.
Does the sequence given by
${a}^{n}$
converge? If so, what is the limit?
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Let
$a\in \mathbb{R}$
with
$a>0$.
Does the sequence given by
${a}^{1/n}$
converge? If so, what is the limit?
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Does the sequence given by
${n}^{1/n}$
converge? If so, what is the limit?
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Let
$a\in \mathbb{R}$
with
$a>0$.
Fix a positive real number
${x}_{1}$. Let
${x}_{n+1}=\frac{1}{2}({x}_{n}+a/{x}_{n})$.
Show that the sequence
${x}_{n}$
converges to
$\sqrt{a}$.
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Let
$\alpha ,\beta \in {\mathbb{R}}_{>0}$.
Let
${a}_{1}=\alpha $ and
${a}_{n+1}=\sqrt{\beta +{a}_{n}}$.
Show that the sequence
${a}_{n}$
converges and find the limit.
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Let
$\alpha ,\beta \in {\mathbb{R}}_{>0}$.
Let
${a}_{1}=\alpha $ and
${a}_{n+1}=\beta +\sqrt{{a}_{n}}$.
Show that the sequence
${a}_{n}$
converges and find the limit.
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Let
${x}_{1}=1$ and
${x}_{n+1}=\frac{1}{2+{x}_{n}}$.
Show that the sequence
${x}_{n}$
converges and find the limit.
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Fix a real number
${x}_{1}$ between 0 and 1.
Let
${x}_{n+1}=\frac{1}{7}({x}_{n}^{3}+2)$.
Show that the sequence
${x}_{n}$
converges and that the limit is a soluntion to the equation
${x}^{3}-7x+2=0$. Use this observation to estimate the solution to
${x}^{3}-7x+2=0$ to three decimal places.
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Find the upper and lower limits of the sequence
${\left(-1\right)}^{n}(1+\frac{1}{n})$.
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Find the upper and lower limits of the sequence given by
${a}_{1}=0$,
${a}_{2k}=\frac{1}{2}{a}_{2k+1}$,
and
${a}_{2k+1}=\frac{1}{2}+{a}_{2k}$. | |

Give an example of a sequence
$\left({a}_{n}\right)$
such that none of
$\mathrm{inf}{a}_{n}$,
$\mathrm{lim}\mathrm{inf}{a}_{n}$,
$\mathrm{lim}\mathrm{sup}{a}_{n}$, and
$\mathrm{sup}{a}_{n}$ are equal.
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Let
${a}_{n}$
be a bounded sequence. Show that
$\mathrm{lim}\mathrm{inf}{a}_{n}\le \mathrm{lim}\mathrm{sup}{a}_{n}$.
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Let
${a}_{n}$
be a bounded sequence. Show that
${a}_{n}$
converges if and only if
$\mathrm{lim}\mathrm{sup}{a}_{n}\le \mathrm{lim}\mathrm{inf}{a}_{n}$.
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Let
${a}_{n}$
be a bounded sequence such that
$\mathrm{lim}\mathrm{sup}{a}_{n}\le \mathrm{lim}\mathrm{inf}{a}_{n}$.
Show that
$\mathrm{lim}\mathrm{sup}{a}_{n}=\mathrm{lim}\mathrm{inf}{a}_{n}=\mathrm{lim}{a}_{n}$.
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Let
${a}_{n}$
be a real sequence.
Show that
${\mathrm{lim}}_{n\to \infty}{a}_{n}=a$ if and only if
$\mathrm{lim}\mathrm{sup}{a}_{n}=\mathrm{lim}\mathrm{inf}{a}_{n}=a$.
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Prove that
$\sum _{k=1}^{n}\frac{1}{k(k+1)}=\frac{n}{n+1}$. |

[BG]
A. Braverman and
D. Gaitsgory,
* Crystals via the affine Grassmanian*,
Duke Math. J.
**107** no. 3, (2001), 561-575;
arXiv:math/9909077v2,
MR1828302 (2002e:20083)