Function convergence's equivalence to sequence convergence
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( E \) be a subset of \( X \), let \( x_0 \) be an adherent point of \( E \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real. Then the following two statements are logically equivalent:
- \( f \) converges to \( L \) at \( x_0 \) in \( E \).
- For every sequence \( (a_n)_{n=0}^{\infty} \) which consists entirely of elements of \( E \) and converges to \( x_0 \), the sequence \( (f(a_n))_{n=0}^{\infty} \) converges to \( L \).
In view of the above proposition, we will sometimes write "\( f(x) \to L\) as \( x \to x_0 \) in \( E \)" or "\( f \) has a limit \( L \) at \( x_0 \) in E" instead of "\( f \) converges to \( L \) at \( x_0 \) in \( E \)" or \( \lim_{x\rightarrow x_0;x\in E}^{\infty} f(a_n) = L \).
Use limits of sequences to calculate limits of functions
This proposition is very important in allowing us to determine what real a function converges to without having to manipulate the definition of function convergence and play around with ε-closeness. Instead we can use all of our built up knowledge of limits of sequences.