HelmutKnaust: Created page with "'''Problem 11.''' Suppose $(a_n)$ is a Cauchy sequence, and that $(b_n)$ is a sequence satisfying $\lim_{n\to\infty} |a_n-b_n|=0$. Show that $(b_n)$ is a Cauchy sequence. '''..."

2025-09-30T15:37:11Z

Created page with "'''Problem 11.''' Suppose $(a_n)$ is a Cauchy sequence, and that $(b_n)$ is a sequence satisfying $\lim_{n\to\infty} |a_n-b_n|=0$. Show that $(b_n)$ is a Cauchy sequence. '''..."

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'''Problem 11.''' Suppose $(a_n)$ is a Cauchy sequence, and that $(b_n)$ is a sequence satisfying $\lim_{n\to\infty} |a_n-b_n|=0$. Show that $(b_n)$ is a Cauchy sequence.

'''Problem 12.''' Suppose $(a_n)$ is a '''bounded''' sequence such that all of its '''converging''' subsequences converge to the same limit, say $L$. Show that $(a_n)$ converges to $L$ as well.

'''Problem 13.''' Consider the following two properties:
# Every non-empty set that is bounded from above has a supremum.
# Every Cauchy sequence converges.
Show that (2)$\Rightarrow$(1). ((1)$\Rightarrow$(2) was done in class, via the Bolzano-Weierstrass Theorem.)

'''Problem 14.'''
# Show: If $x$ is an accumulation point of $A\cup B$, then $x$ is an accumulation point of $A$, or $x$ is an accumulation point of $B$ (or both).
# Does the result also hold for a countably infinite collection of sets? Give a proof, or provide a counterexample.

'''Problem 15.''' Find all accumulation points of the set
\[\left\{\frac{1}{m}+\frac{1}{n}\ |\ m,n\in\mathbb{N}\right\}\]
Remember that $A=B\ \Leftrightarrow\ (A\subseteq B)\wedge (B\subseteq A)$.

CRN 10459: HW 3 - Revision history

HelmutKnaust: Created page with "'''Problem 11.''' Suppose $(a_n)$ is a Cauchy sequence, and that $(b_n)$ is a sequence satisfying $\lim_{n\to\infty} |a_n-b_n|=0$. Show that $(b_n)$ is a Cauchy sequence. '''..."