How can I code the equations with big brackets on the right and left sides shown in the picture? Thank you for your help!
[Tex/LaTex] Latex bracket on multiple lines
brackets
Related Solutions
The mtpro2 package (MathTime Professional II) provides extra-large (up to 10 cm tall) fence symbols -- round parentheses, square brackets, curly braces, angle brackets, etc. -- via its \LEFTRIGHT command. Note that the full mtpro2 package isn't free of charge; however, its "lite" subset, which is all that's needed to produce extra-large fence symbols, is free of charge.
\documentclass{article}
\usepackage[lite]{mtpro2}
\begin{document}
\[
\LEFTRIGHT\langle\rangle{
\begin{array}{c}
1\\2\\3\\4\\5\\6\\7\\8\\9\\10
\end{array}}
\]
\end{document}
update 2020-10-01: The pctex.com site noted in my answer no longer seems provide a functioning link to the page from which the lite subset of the mtpro2 package used to be available for downloading. Many thanks to @mateuszb for bringing this issue to my attention.
You need the rcases or drcases environment from mathtools:
\documentclass{article}
\usepackage{mathtools}
\begin{document}
Jede Teilfolge hat entweder unendliche viele Elemente von $(g_{2n})$ oder
$(g_{2n+1})$.
\[
\begin{drcases}
g_+ = \lim a_n = \lim g_{2n} = g\\
g_+ = \lim b_n = \lim g_{2n} = g
\end{drcases}
g_+ = g_- \Rightarrow \lim g_n = g = g_- = g_+ = \frac{1 + \sqrt{5}}{2}
\]
\end{document}
Note that you're using too many braces; also there's no need to have \left or \right before all delimiters: it's even discouraged, as it affects spacing.
Usually I also recommend to always have braces around subscripts and superscripts, even if they are only one token, but an experienced user knows when to sin against this “rule”.
I wouldn't use brackets around the whole construction, but the document is yours; here is how you can do it:
\documentclass{article}
\usepackage{mathtools}
\usepackage{lipsum} % just for the example
\newsavebox{\tempbox}
\begin{document}
\lipsum*[2]
\[
\sbox{\tempbox}{%
$\displaystyle
\begin{drcases}
g_+ = \lim {a_n} = \lim g_{2n} = g\\
g_+ = \lim {b_n} = \lim g_{2n} = g
\end{drcases}
g_+ = g_- \Rightarrow \lim g_n = g = g_- = g_+ = \frac{1 + \sqrt{5}}{2}
$%
}
\begin{pmatrix}
\,\parbox[b]{\wd\tempbox}{
Jede Teilfolge hat entweder unendliche viele Elemente von
$(g_{2n})$ oder $(g_{2n+1})$.
}\hfill\, % left alignment
\\[\abovedisplayskip]
\,\usebox{\tempbox}\,
\end{pmatrix}
\]
\lipsum[3]
\end{document}
Best Answer
This works because
stackenginecan set equations with fixed baselineskip.