add further edges

nodes/114/114.tex

@@ -11,7 +11,8 @@
 \end{Definition}
 
 We make essential use of Dedekind's theorem to prove the following result:
-\begin{Theorem}[Convergence of bounded and monotonic sequences]\label{thm:monbndseq}
+\begin{Theorem}[Convergence of bounded and monotonic sequences]
+ \label{thm:monbndseq}
  Let $(a_n)_{n\in\mathbb{N}}$ be a~real sequence that has one of the following properties:
 \begin{itemize}
  \item[--] $(a_n)_{n\in\mathbb{N}}$ is monotonically increasing and bounded from above;
@@ -47,7 +48,7 @@ Thus, $(a_n)$ is bounded from below. For showing monotonicity, we consider
 \[a_{n+1}-a_n=\frac{a_n+\frac2{a_n}}{2}-a_n=\frac{1}{2a_n}(2-a_n^2).\]
 In particular, if $n\geq2$, we have that $a_n>0$ and $2-a_n^2\leq0$. Thus, $a_{n+1}-a_n\leq0$ for $n\geq2$. An~application of Theorem~\ref{thm:bndmonseq} (resp.\ the slight generalisation in Remark %\ref{rem:monseqgen}
 from above) 
-now leads to the existence of some $a\in\mathbb{R}$ with $a=\lim_{n	o\infty}a_n$.\\[2ex]
+now leads to the existence of some $a\in\mathbb{R}$ with $a=\lim_{n	\to\infty}a_n$.\\[2ex]
 To compute the limit, we make use of the relation $\lim_{n \to\infty}a_n=\lim_{n\to\infty}a_{n+1}$ (follows directly from the Definition of limits) and the formulae for limits. This yields
 \[a=\lim_{n \to\infty}a_n=\lim_{n\to\infty}a_{n+1}=\lim_{n\to\infty}\frac{a_n+\frac2{a_n}}{2}=\frac{a+\frac2{a}}{2}.\]
 This relation leads to the equation $2-a^2=0$, i.e., we either have $a=\sqrt{2}$ or $a=-\sqrt{2}$. However, the latter solution cannot be a limit since all sequence elements are positive. Therefore, we have

nodes/cyto.json

@@ -310,7 +310,7 @@
     },
     "114": {
       "id": "114",
-      "label": "Convergence of Bounded Monotonic Sequences",
+      "label": "Bounded Monotonic\nSequences",
       "meta": " ",
       "content": "If a sequence of real numbers is bounded and monotonic, then it is convergent.",
       "notes": "114-snippet.html",
@@ -1889,6 +1889,31 @@
       "target": "104",
       "label": "The proof and a lot of applications of the sandwich theorem are only possible because the limit theorems allow for elementary calculations with limits."
     },
+    "101-114": {
+      "source": "101",
+      "target": "114",
+      "label": "If a sequence is bounded and monotonic, then it converges."
+    },
+    "102-114": {
+      "source": "102",
+      "target": "114",
+      "label": "Boundedness alone does not give convergence yet, one also needs monotonicity."
+    },
+    "106-114": {
+      "source": "106",
+      "target": "114",
+      "label": "The convergence of bounded sequences follows from Dedekind's theorem which guarantees the existence of suprema of bounded sets."
+    },
+    "105-114": {
+      "source": "106",
+      "target": "114",
+      "label": "The limit of a monotonic increasing and bounded sequence is precisely the supremum of its function values. An analogous connection holds for the infimum."
+    },
+    "013-102": {
+      "source": "013",
+      "target": "102",
+      "label": "The image of a sequence, i.e. the set of all values that the sequence attains, is a bounded set."
+    },
     "203-207": {
       "source": "203",
       "target": "207",