So according to sources such as MathWorld and Wikipedia, as well as various Real Analysis textbooks, a sequence is defined as divergent if it is not convergent.
But here's an interesting problem...
Prove that the sequence $ \{x_n\} = \{(-1)^n\} $ is divergent.
Solution:
The sequence $ \{x_n\} $ is bounded (we can take $ M = 1 $), which prevents us from directly applying the convergence theorem that states every convergent sequence is bounded.
Assume $ a = \lim_{n \to \infty} x_n $ exists. Let $ \epsilon = 1 $. Then, there exists a natural number $ K_1 $ such that:
$$| (-1)^n - a | < 1 \quad \text{for all } n \geq K_1.$$
For odd $ n $ (where $ n \geq K_1 $), this implies:
$$| -1 - a | < 1 \quad \Rightarrow \quad -2 < a < 0.$$
For even $ n $ (where $ n \geq K_1 $), we have:
$$| 1 - a | < 1 \quad \Rightarrow \quad 0 < a < 2.$$
Since $ a $ cannot satisfy both inequalities simultaneously, our assumption that $ \{x_n\} $ converges leads to a contradiction. Therefore, the sequence $ \{x_n\} $ is divergent.
But here's an interesting take...
According to the course book I am currently reading and again many other Real Analysis books and Encyclopaedia of Mathematics, a divergent sequence is defined as one for which
$$\lim_{n \to \infty} x_n = \infty \quad \text{or} \quad \lim_{n \to \infty} x_n = -\infty.$$
Both definitions can be connected in a way or other but we have something else to handle currently.
Since
$$\lim_{n \to \infty} (-1)^n$$
is indeterminate, the sequence $ \{x_n\} = \{(-1)^n\} $ is neither convergent nor divergent according to this definition.
This discrepancy raises confusion regarding which definition is more appropriate. If the latter definition holds, then the initial problem makes no sense.
Why is there no, one concrete definition about this topic in mathematics? Different sources and different books gives different definitions. This ambiguity is perplexing, especially since I have not encountered similar issues previously. Which approach should I adopt, and which definition is ultimately correct (or more correct)? I understand that there can be multiple ways to prove or disprove a mathematical statement, which is why I'm seeking clarification on this problem. As I'm new to Real Analysis, I am not aware of any other approaches through which I can try to solve this problem.