I'm proving that if a sequence $\{x_n\}$ has the property
\[
\lim_{n\rightarrow\infty}x_{2n}=\lim_{n\rightarrow\infty}x_{2n+1}=\infty
\]
show that the sequence $\{x_n\}$ diverges to $\infty$.
My method is to let $M\in\mathbb{R}$, then we know from above that
\begin{align*}
&\exists N_1\in\mathbb{N},\forall n\geq N_1, x_{2n}\geq M \\
&\exists N_2\in\mathbb{N},\forall n\geq N_2, x_{2n+1}\geq M
\end{align*}
Let $N=\max\{N_1,N_2\}$ and consider $n\geq N$. Now, you could use modular arithmetic to show the theorem. However, my professor suggested it was possible to select a max for a $N$ that sh…
\[
\lim_{n\rightarrow\infty}x_{2n}=\lim_{n\rightarrow\infty}x_{2n+1}=\infty
\]
show that the sequence $\{x_n\}$ diverges to $\infty$.
My method is to let $M\in\mathbb{R}$, then we know from above that
\begin{align*}
&\exists N_1\in\mathbb{N},\forall n\geq N_1, x_{2n}\geq M \\
&\exists N_2\in\mathbb{N},\forall n\geq N_2, x_{2n+1}\geq M
\end{align*}
Let $N=\max\{N_1,N_2\}$ and consider $n\geq N$. Now, you could use modular arithmetic to show the theorem. However, my professor suggested it was possible to select a max for a $N$ that sh…
I am starting to learn some measure theory, and I was wondering if there is a name for $\mathfrak{M}$.
I have the definition:
A collection $\mathfrak M$ of subsets of a set $X$ is said to be a $\sigma$-algebra in $X$ if $\mathfrak M$ has the following properties:
blahblah
If $\mathfra...
