\input /home/pasky/school/fastex/lib.tex

\subchapter{Limesy}{

Mějme $\{a_{n}\}_{n\in\nat}$. Definujme posloupnosti $\{b_{k}\}_{k\in\nat}$, $\{c_{k}\}_{k\in\nat}$:
	$$ b_{k} = \sup \{ a_{n}: n \ge k,\ n \in \nat \} $$
	$$ c_{k} = \inf \{ a_{n}: n \ge k,\ n \in \nat \} $$


\scpart{Pozorování:}{
	$$ c_{k} \le a_{k} \le b_{k}\qquad \forall{k\in\nat} $$
	$$ \{b_{k}\}_{k\in\nat} \textbox{je \bf nerostoucí} \Rightarrow{} \existss \lim_{k\to\infty} b_{k} $$
	$$ \{c_{k}\}_{k\in\nat} \textbox{je \bf neklesající} \Rightarrow{} \existss \lim_{k\to\infty} c_{k} $$
}


\scpart{Limes superior}{
	$$ \limsup_{n\to\infty} a_{n} \defined \cases{
					    \displaystyle{\lim_{k\to\infty} b_{k}}&$\lim b_{k} \in \real$\cr
	                                    +\infty                &$\{a_{n}\}$ je shora neomezená
					    } $$
}

\scpart{Limes inferior}{
	$$ \liminf_{n\to\infty} a_{n} \defined \cases{
					    \displaystyle{\lim_{k\to\infty} c_{k}}&$\lim c_{k} \in \real$\cr
	                                    -\infty                &$\{a_{n}\}$ je zdola neomezená
					    } $$
}


\example{
	\list{
		\listItem{ $ a_{n} = (-1)^n$, $\liminf a_{n} = -1$, $\limsup a_{n} = 1 $}
		\listItem{ $ a_{n} = 1/{n^2}$, $\liminf a_{n} = \limsup a_{n} = \lim a_{n} = 0 $}
	}
}


\notes{
	\list{
		\listItem{ Narozdíl od limity, která nemusí existovat, $\liminf a_{n}$ a $\limsup a_{n}$ existují vždy.}
		\listItem{ $ \liminf a_{n} \le \limsup a_{n} $}
	}
}


\theorem{10}{o vztahu limity, suprema a limes inferior}{

	$$ \lim_{n\to\infty} a_{n} = A \in \real^*  \Longleftrightarrow  \limsup_{n\to\infty} a_{n} = \liminf_{n\to\infty} a_{n} = A \in \real^* $$



\proof{
	\proofleftimpl{

		Víme: $ \forall k\in\nat :\ c_{k} \le a_{k} \le b_{k} $

		$$ \lim_{k\to\infty} c_{k} = \lim_{k\to\infty} b_{k} = A \in \real^*
		   \becauseof{2 policajti}{\Longrightarrow} \lim_{k\to\infty} a_{k} = A $$

	}

\penalty-100
	\proofrightimpl{
	\list{

		\listItem{Nechť $A\in\real$:

			Zvol $\eps > 0$. Potom $\existss n_{0}$ takové, že $\forall n \ge n_{0}$:
			$$ |a_{n}-A| < \eps $$
			$$ A - \eps \le a_{n} \le A + \eps $$
			$$ A - \eps \le c_{n} \le a_{n} \le b_{n} \le A + \eps $$
			$$ 0 \le b_{n} - c_{n} \le 2\eps\qquad (\forall{n \ge{} n_{0}}) $$
			$$ \Rightarrow \lim_{n\to\infty} (b_{n}-c_{n}) = 0
			       \Rightarrow \lim_{n\to\infty} b_{n} = \lim_{n\to\infty} c_{n} $$
		}

		\listItem{Nechť $A = +\infty$:

			Pak $\{a_{n}\}$ je shora neomezená.
			$$ \limsup_{n\to\infty} a_{n} = +\infty $$

			Zvolíme $K \in \real$. Pak $\existss n_{0}\in\nat,\ \forall n \ge n_{0} :\ a_{n} > K$.
			$$ \inf_{n \ge n_{0}} \{a_{n}\} \ge K  \Rightarrow  c_{n_{0}} \ge K $$

			Protože $\{c_{k}\}$ je neklesající, máme $\forall n \ge n_{0} :\ c_{n} \ge K$.
			$$ \lim_{n\to\infty} c_{n} = +\infty $$
		}

		\listItem{Nechť $A = -\infty$: analogicky.}

	}
}

}


\exercise{
	Dokažte:
	$$ \liminf a_{n} + \liminf b_{n} \le \liminf(a_{n} + b_{n}) \le \limsup(a_{n} + b_{n}) \le \limsup a_{n} + \limsup b_{n} $$
}

}

\bend