Question about the definition of a limit of sequence of sets Let $X$ be any set and $(A_n)$ be sequence of subsets of $X$. MY book defines 
$$ \lim_{n \to \infty} A_n = \bigcup_{n \geq 1 } A_n $$
if $A_n \subset A_{n+1}$ and 
$$ \lim_{n \to \infty} A_n = \bigcap_{n \geq 1} A_n $$
if $A_{n} \supset A_{n+1} $.
But, my teacher defines the limit as follows: $A_n $ converges to $A$ if 
$$ \lim_{ n \to \infty} \chi_{A_n} = \chi_A $$
Are theu equivalent definitions? 
 A: First of all, the formal definition that I have seen most in literature is the following:
We define 
$\limsup{A_n}:=\bigcap_{n\in\mathbb{N}}\bigcup_{k\geq n}A_k$ 
$\liminf{A_n}:=\bigcup_{n\in\mathbb{N}}\bigcap_{k\geq n}A_k$
Meaning that $\limsup{A_n}:=\{x: x\in A_n \textrm{ for an infinite number of }n\}$, $\liminf{A_n}:=\{x: x\in A_n \textrm{ for almost all }n\}$.
Then we go on to define that a limit of a sequence of sets exists if and only if $\limsup{A_n}=\liminf{A_n}$.
I'd like to note here that, yes, in the examples in your book those two definitions yield the same. However, I do not like the phrasing very much since in your case the definition is different depending on the actual sequence. I'd think it'd be a lot more straightforward to say that if $A_n$, is increasing, then $\lim A_n=\bigcup_n A_n$, not the other way around.
Now, for the second definition. I assume you are speaking about pointwise convergence on the space $\Omega^{\{0,1\}}$, where $\Omega$ is the 'largest' set, where all your sets are defined, since given $A_n=\{k\in\mathbb{N},k\leq n\}$, you will never reach uniform convergence, and you cannot even begin to talk about anything resembling continuity here.
Given the above definition, if a limit $A$ exists, then each point $x\in A$ must be in $A_n$ for almost all $n$. Hence there is a $m$, such that $x\in A_n\quad\forall n>m$. 
Given any point $y\notin A$, we know that $y$ is not in $A_n$ for an infinite number of $n$, hence there is a $m$ such that $y\notin A_n\quad\forall n>m$. 
Those two properties say that if we have a limit $A$, then our sequence $\chi_{A_n}$ converges pointwise to $\chi_A$. 
With the very same argument we can go the way back - if our sequence $\chi_{A_n}$ converges to $\chi_A$, then for each point $x\in A$ we know that there is a $m$ as above, meaning $x$ is in almost all $A_n$, meaning $x\in\liminf{A_n}\subseteq\limsup{A_n}$. 
Now, given an arbitrary $x\in\limsup{A_n}$, we know that it must be in $A$, since otherwise $\chi_{A_n}\nrightarrow \chi_A$ - hence we have $A\subseteq\liminf_{A_n}\subseteq\limsup_{A_n}\subseteq A \Rightarrow A=\liminf_{A_n}=\limsup_{A_n}=\lim_{A_n}$.
So, in fact, the latter definition is equivalent with the one I know from literature. 
For the two concrete examples you stated, those two are the very same. Seeing as the one given by your teacher is much more general, I personally prefer this one. However, the other one might be a bit more intuitive and in these special cases easier to work with. I'd suggest you remember both of them, get used to both of them and apply them as necessary - tasks from the book you should be able to solve with either methods, those given by your teacher I'd try to solve given his method. They are, in these special cases, equivalent definitions, but the latter is just a lot more general.
