Q3Mathematics I

In the strict mathematical framework of real analysis, a sequence of real numbers, denoted as $\{a_n\}$, is formally said to be convergent if, as the index $n$ gets infinitely large ($n\to \infty$), the terms of the sequence strictly approach a single, unique, and absolute finite real number $L$. This fixed target number $L$ is officially known as the limit of the sequence.

Mathematically, this strict convergence is concisely expressed using the standard limit notation:

According to Cauchy's rigorous $(ϵ-N)$ definition, this means that for any arbitrarily small positive number $ϵ>0$, there definitively exists a corresponding positive integer $N$ such that the absolute difference $|a_n-L|<ϵ$ for all index values $n>N$. If a sequence fails to approach a finite, unique number (for example, if it grows to infinity or violently oscillates without settling), it is rigorously classified as divergent.