59 lines
1.1 KiB
TeX
59 lines
1.1 KiB
TeX
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\documentclass{article}
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\input{../../common/preamble}
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\begin{document}
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\begin{xtheorem}{I.27}
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Every nonempty set $S$ that is bounded below has a greatest lower bound;
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that is, there is a real number $L$ such that $L = \inf{S}$.
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_isGLB}
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\end{proof}
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\begin{xtheorem}{I.29}
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For every real $x$ there exists a positive integer $n$ such that $n > x$.
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_pnat_geq_self}
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\end{proof}
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\begin{xtheorem}{I.30}[Archimedean Property of the Reals]
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If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive integer $n$ such that $nx > y$.
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_pnat_mul_self_geq_of_pos}
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\end{proof}
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\begin{xtheorem}{I.31}
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If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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$$a \leq x \leq a + \frac{y}{n}$$
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for every integer $n \geq 1$, then $x = a$.
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.forall_pnat_leq_self_leq_frac_imp_eq}
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\end{proof}
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\end{document}
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