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Created page with "'''Problem 11.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a '''decreasing''' sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges ..."

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'''Problem 11.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a '''decreasing''' sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$.

'''Problem 12.''' 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 13.''' Let $(a_n)_{n=1}^\infty$ be a Cauchy sequence, and let $\varphi:\mathbf{N}\to\mathbf{N}$ be a one-to-one function. Show that the sequence $(a_{\varphi(n)})_{n=1}^\infty$ is a Cauchy sequence.

'''Problem 14.''' 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 15.''' 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.)

CRN 11247: HW 3 - Revision history

HelmutKnaust: Created page with "'''Problem 11.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a '''decreasing''' sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges ..."