Definition: A sequence \( \{a_n\} \) converges to \( L \) if for every \( \epsilon > 0 \), there exists \( N \in \mathbb{N} \) such that \( |a_n - L| < \epsilon \) for all \( n \geq N \).
In real analysis, sequence convergence is fundamental. It describes the behavior of a sequence approaching a limit as the number of terms increases. This concept forms the foundation for understanding limits, continuity, and differentiability in calculus.
Definition: A function \( f \) is continuous at a point \( c \) if for every \( \epsilon > 0 \), there exists \( \delta > 0 \) such that if \( |x - c| < \delta \), then \( |f(x) - f(c)| < \epsilon \).
The definition of continuity means that for a function to be continuous at a point, small changes in the input around that point must result in small changes in the output. This concept is crucial for understanding the behavior of functions and their graphs, especially in calculus. For example: