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nodes/204/204.tex

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 \begin{Theorem}[Cauchy Criterion]
-A~series $\displaystyle\sum_{k=1}^\infty a_k$ in $\K$ is convergent if and only if for all $\varepsilon>0$, there exists some $N$ such that for all $n\geq m\geq N$ holds
+A~series $\displaystyle\sum_{k=1}^\infty a_k$ in $\mathbb{R}$ is convergent if and only if for all $\varepsilon>0$, there exists some $N$ such that for all $n\geq m\geq N$ holds
 \[\left|\sum_{k=m}^n a_k\right|<\varepsilon.\]
 \end{Theorem}
 
 {\em Proof:} By the theorems on completeness and convergence of cauchy sequences, a~series converges if and only if the sequence $(s_{n})_{n\in\mathbb{N}}$ of partial sums is a~Cauchy sequence.\\
 On the other hand, for $n\geq m$, we have
 \[\left|s_n-s_{m-1}\right|=\left|\sum_{k=m}^n a_k\right|.\]
-Therefore, the Cauchy criterion is really equivalent to the fact that $(s_{n})_{n\in\mathbb{N}}$ is a~Cauchy sequence in $\K$.\hfill$\Box$
+Therefore, the Cauchy criterion is really equivalent to the fact that $(s_{n})_{n\in\mathbb{N}}$ is a~Cauchy sequence in $\mathbb{R}$.\hfill$\Box$
 
-\begin{Remark}{}
-%For incomplete spaces, the Cauchy criterion is only necessary (but not sufficient) for convergence of a~series. This is a~consequence of the fact that %the Cauchy criterion is equivalent to the fact that
-%the sequence of partial sums is a~Cauchy sequence.\\
-Reconsidering the example at the very beginning of this chapter, the divergence of this sequence can be directly verified be employing the Cauchy criterion.
-\end{Remark}
+%  \begin{Remark}{}
+%  %For incomplete spaces, the Cauchy criterion is only necessary (but not sufficient) for convergence of a~series. This is a~consequence of the fact that %the Cauchy criterion is equivalent to the fact that
+%  %the sequence of partial sums is a~Cauchy sequence.\\
+%  Reconsidering the example at the very beginning of this chapter, the divergence of this sequence can be directly verified be employing the Cauchy criterion.
+%  \end{Remark}
 As a corollary, we can formulate the following criterion.
 
 \begin{Theorem}[Necessary criterion for convergence of series]
 \label{eq:conv0}
-Let \[\sum_{k=1}^\infty a_k\] be a~convergent series in $\K$. Then $(a_n)_{n \in \mathbb{N}}$ is convergent with
+Let \[\sum_{k=1}^\infty a_k\] be a~convergent series in $\mathbb{R}$. Then $(a_n)_{n \in \mathbb{N}}$ is convergent with
 \[\lim_{n\to\infty}a_n=0.\]
 \end{Theorem}
 {\em Proof:}