- #1
darkestar
- 12
- 0
Hello!
I just finished typing up my first Latex document for a proof I worked on. Now, I'm having trouble posting it on these forums. Here is the source code...
\begin{document}
$f : \mathbb{R} \Rightarrow \mathbb{R}$ is odd $\;\Longleftrightarrow \;f(-x) = -f(x) \;\forall x$.
Show that if $f : \mathbb{R} \Rightarrow \mathbb{R}$ is odd and with a retriction to $[0, \infty)$ is strictly increasing, then $f : \mathbb{R} \Rightarrow \mathbb{R}$ is stricly increasing.
\begin{proof}
Suppose $f : \mathbb{R} \Rightarrow \mathbb{R}$ is odd and strictly increasing on $[0, \infty)$. Consider $x_n, x_{n+1} \in [0, \infty)$, where $n \in \mathbb{N}$ and $x_{n+1} > x_n$. Since $f$ is strictly increasing in $[0, \infty)$, then $f(x_{n+1}) > f(x_n)$. Additionally, since $f$ is odd, $f(-x_n) = -f(x_n)$ and $f(-x_{n+1}) = -f(x_{n+1})$. Thus,
\begin{align*}
\\f(x_{n+1}) > f(x_n)
\\\Rightarrow -f(x_n) > -f(x_{n+1})
\\\Rightarrow f(-x_n) > f(-x_{n+1})
\\\Rightarrow -x_n > -x_{n+1}
\end{align*}
\\Therefore, $f$ is strictly increasing on $(-\infty, 0) \cup [0, \infty)$. Hence, $f : \mathbb{R} \Rightarrow \mathbb{R}$ is strictly increasing.
\end{proof}
\end{document}
It compiles fine using TeXWorks, although I'm not sure how to modify it so I can post it here. Thanks for the help!
I just finished typing up my first Latex document for a proof I worked on. Now, I'm having trouble posting it on these forums. Here is the source code...
\begin{document}
$f : \mathbb{R} \Rightarrow \mathbb{R}$ is odd $\;\Longleftrightarrow \;f(-x) = -f(x) \;\forall x$.
Show that if $f : \mathbb{R} \Rightarrow \mathbb{R}$ is odd and with a retriction to $[0, \infty)$ is strictly increasing, then $f : \mathbb{R} \Rightarrow \mathbb{R}$ is stricly increasing.
\begin{proof}
Suppose $f : \mathbb{R} \Rightarrow \mathbb{R}$ is odd and strictly increasing on $[0, \infty)$. Consider $x_n, x_{n+1} \in [0, \infty)$, where $n \in \mathbb{N}$ and $x_{n+1} > x_n$. Since $f$ is strictly increasing in $[0, \infty)$, then $f(x_{n+1}) > f(x_n)$. Additionally, since $f$ is odd, $f(-x_n) = -f(x_n)$ and $f(-x_{n+1}) = -f(x_{n+1})$. Thus,
\begin{align*}
\\f(x_{n+1}) > f(x_n)
\\\Rightarrow -f(x_n) > -f(x_{n+1})
\\\Rightarrow f(-x_n) > f(-x_{n+1})
\\\Rightarrow -x_n > -x_{n+1}
\end{align*}
\\Therefore, $f$ is strictly increasing on $(-\infty, 0) \cup [0, \infty)$. Hence, $f : \mathbb{R} \Rightarrow \mathbb{R}$ is strictly increasing.
\end{proof}
\end{document}
It compiles fine using TeXWorks, although I'm not sure how to modify it so I can post it here. Thanks for the help!
