Does the following fail to qualify as a sequence?
$\{x_n\}_{n=1}^\infty$ such that $x_n= \frac{1}{n-2}$
I ask because Basic Analysis by Lebl gives the following definition: "A sequence (of real numbers) is a function x: $\mathbb N \rightarrow \mathbb R$." By my understanding, my example meets this definition.
Yet I also see the theorem "a convergent sequence is bounded." My example seems to be convergent (the limit equals zero), but I don't think it's bounded because of when $n = 2$.
A follow up question here: many of you have pointed out that my function is undefined when $ n = 2 $. But what about the sequence $ a_n = n $? In what sense is this defined as $ n \rightarrow \infty $ ? I intuitively understand there's a distinction, but it would be hard for me to rigorously express what the distinction is.
Best Answer
It does not qualify (in regards to the definition you mentioned).
Let's take a step back. You see, the definition of a sequence changes depending on which perspective (book, course, etc.) you take. Let's define a sequence to be, say, a map from $\{1,2,3,4\}$, the index set, to the English alphabet. $\{M,A,T,H\}$ is therefore a sequence by our definition. Of course, this is very loose, but essentially, Mathematics originated from different places around the world at some point in time, but today we have an "international" language that fails to regard the exact meanings of certain words. This is because we can only have a finite amount of words in our language, and it is a very very small finite number. Thus, a widespread definition of the word sequence (which is a relatively common word) would not help to summarize the thousands that have been created across time.
By definition, $\{x_n\}^{\infty}_{n=1}$ for $x_n=\frac{1}{n-2}$ is a map $M:\mathbb{N}\backslash\{2\}\mapsto\mathbb{R}$, which does not follow the presented definition of a sequence $S:\mathbb{N}\mapsto\mathbb{R}$. Thus, it can be concluded that $x_n$ is not a sequence.
Edit: I wanted to answer your follow-up question that you added. The sequence $a_n$ is "defined" as $n$ approaches infinity, but not at infinity, because infinity is not in the domain of $\mathbb{N}$ nor $\mathbb{R}$. However, we can define the word approaches in such a way that it means the end behavior of a function. Thus, the sequence $a_n=n$ approaches $\infty$ as $n$ approaches $\infty$. But never will a sequence with a mapping from the natural numbers to the reals have a mapping when $n=\infty$ because infinity is in neither of the domains of the image and the pre-image under the map. Approaching something is a concept that can be explained rigorously, nothing more. You don't need to know these proofs for an introduction, only know that "approaching", for now at least, means the end behavior of such sequences.