Must a sequence be a function on $\mathbb{N}$? I saw many books giving definition of a sequence as:

A sequence $\left\{a_n\right\}$ is a function, $a:\mathbb{N}\to \mathbb{R}.$

This means that the domain is the set of natural numbers. But there are some sequences for which the first few natural numbers are not the part of the domain.  An example is
$$a_n=\frac{1}{n^2-n}.$$
So will it be more better if we define the sequence as, a function $$a:A \to \mathbb{R},$$ where $A \subseteq \mathbb{N}$.
 A: Something to always be aware of in mathematics is that definitions are not gospel, they are not written in stone, and they are not handed down by g*d or nature.  The particular definitions we choose are those definitions which we find to be useful, and which describe objects which we might like to study.
In the case of sequences, the fundamental underlying intuition is that we have a bunch of objects which occur in succession, one after another.  A sequence has a starting term, and then a next term, and then a next term, and so on, forever and ever.[1]  This fundamental intuition leads us to compare an ordered list of numbers (or other objects) to the natural numbers, which are, in some sense, the simplest collection of objects which match the description "there is a first thing, and then other things which occur in succession".
In other words, we can take any sequence and match up the objects in the sequence to natural numbers.  In the modern language of mathematics, which uses functions to do a lot of the heavy lifting, this "naturally" leads to the following definition:

Definition: A sequence of real numbers is a function $a : \mathbb{N}\to\mathbb{R}$.[2]

This is a perfectly good definition for the purposes of most introductory calculus or analysis classes, but, as noted in the question, there are potential problems.  The difficulty in the question is that the asker wants to define a sequence $(a_n)$ via the formula
$$ a_n = \frac{1}{n^2 - n}. $$
Per the definition given above, this is not a sequence, as this formula does not define a function whose domain is $\mathbb{N}$.  That being said, a reasonable understanding of this formula is that the author wishes to define a sequence whose first several terms are
$$ \frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \dotsc.$$
We could say that this "sequence" is defined by the formula above, and concede that it isn't really a sequence because $a_1$ is undefined.  Or we could introduce the notion of a partial function, and define sequences to be certain kinds of partial functions defined on $\mathbb{N}$.  Or we could write another, more general definition of a sequence which has as its domain some singly-generated inductive set. Or we could reindex things and define a sequence $(a'_n)$ by the formula
$$ a'_n = \frac{1}{(n+1)^2 - (n+1)}. $$
Or we can just sweep all of this under the rug and ignore the problems with this "sequence", and get on with doing whatever computation we are actually interested in.
Most of the time, we ignore these issues.  This is not an unreasonable thing to do, as the definition was written to describe a particular intuition regarding what a sequence should be.  We (hopefully) infer that the formula
$$ a_n = \frac{1}{n^2 - n}$$
is meant to define a function $a : \mathbb{N}\setminus\{1\} \to \mathbb{R}$, and get on with our lives.
To get some context for this faffery, you might like to read Terrence Tao's essay There's more to mathematics than rigor and proofs.  You are correct that, from a rigorous point of view, there is a problem with the definition provided here.  It is good that you spotted it—this indicates that you are comfortably sitting in the rigorous state of mathematical thinking (at least, with respect to this particular part of mathematics).  The trick, now, is to gain comfort with this definition (with all of its flaws), build the intuition, and transition to a post-rigorous way of thinking.  Good luck. :)

[1] One might argue that the notion of a "finite sequence" might also be a useful object to study, but let us, for the sake of argument, agree that a "finite sequence" isn't a sequence at all.
[2] Note that we could define a sequence more generically as a function $a : \mathbb{N}\to X$, where $X$ is any set at all.  For example, a function $a :\mathbb{N}\to\mathbb{C}$ is a sequence (of complex numbers), and a sequence $a : \mathbb{N} \to \mathbb{R}^n$ is sequence (of $n$-dimensional vectors).
A: The correct formal definition is the one you quote: a function with $\mathbb{N}$ as domain.
In the example you offer, the definition of $a_n$ makes sense only for $n > 1$. In practice that's not a problem. If you insist on writing that sequence using the formal definition, you shift the index and write
$$
a_n = \frac{1}{(n+1)^2 - (n+1)}.
$$
(This assumes the natural numbers start at $1$.)
A: As Xander says, mathematical definitions are not things to be proven. However, I disagree with some points in his post, not because it contains wrong mathematics but because there is a better way from a logic-based viewpoint.
The whole point of the notion of "sequence" is to capture some kind of listing or iteration. So in line with what Xander says, we can encode such an abstract notion via functions on some index set $I$ that are linearly ordered by $<$, using elements from $I$ to tag the items in the sequence to represent their order in the sequence. An item $x$ assigned tag $k∈I$ can of course be encoded as the pair $⟨k,x⟩$, and thus a sequence with index set $I$ can be encoded by a function on $I$.
Note that this is just one possible encoding of the concept of sequences. It is not the only way to encode sequences. After all, a function on $I$ can also be encoded as a certain kind of set of pairs from $I×S$ for some set $S$, and if you combine both encodings you would get an encoding of a sequence as a certain kind of set of pairs.
The point is that we need an encoding of an abstract notion only if that notion is not inbuilt into the foundational system we are working in. And for us to consider an encoding correct, it must satisfy the properties that we want the notion to satisfy. For sequences, we want to be able to construct new sequences, get a given sequence's length $l$ or extract the $k$-th item from it (where $k≤l$), concatenate two given sequences, and so on. Once we establish that some encoding satisfies all the properties we want, then we define "sequence" to be that encoding, allowing us to reason about sequences in the way we want.
Is the encoding the same as the abstract concept itself? Well, no. It is the same thing as with 'the' real numbers. The construction of $ℝ$ via Cauchy sequences or Dedekind cuts are different, but satisfy the same properties we want (i.e. Dedekind-complete ordered field). In short, we only use $ℝ$ via its interface, and every theorem about $ℝ$ is actually about every complete ordered field.
Likewise, sequences on $ℕ$ (i.e. with index set $ℕ$) are different from sequences on $ℕ_{≥3}$ (i.e. with indices being naturals no less than $3$), but have the same properties. That is precisely why we can view things like $(\frac1n)_{n∈ℕ_{>0}}$ as essentially the same sequence as $(\frac1{n+1})_{n∈ℕ}$, even though the two are different encodings. They have the same length, the same items in the same order, and yield the same result when concatenated with a given other sequence.
Now once you hold this general view of a sequence as indexed by some set $I$, it is completely natural to ask what kind of $⟨I,<⟩$ we want to have. While item-extraction and concatenation works for any $⟨I,<⟩$, length does not really make sense unless $⟨I,<⟩$ is a well-order. This automatically yields the fully general definition of a (possibly transfinite) sequence, including finite sequences (since $⟨[1..n],<⟩$ is also a well-order, being a prefix of $⟨ℕ,<⟩$), even the empty sequence (with length $0$), sequences of length $ω$ (i.e. the length of $⟨ℕ,<⟩$, and also sequences of longer lengths.
The reason we want to have sequences longer than $ω$ is that there are some circumstances in mathematics where we need to perform a recursive iteration longer than $ω$. A recursive function on a well-order $⟨I,<⟩$ is a function $f$ on $I$ given by a recursive definition $f(k) = g(f{↾}I_{<k})$ for every $k∈I$, where $g$ is some function. Such recursion is well-defined only when $⟨I,<⟩$ is a well-order. Admittedly, almost all applied mathematics does not require sequences longer than $ω$, but it is quite common in modern mathematics.
A: Real Sequence is a function whose domains is natural number and range is subset real numbers but most important part is it must be successively. So you can even start with 2 or 3 but terms need to go successively upto infinity.
