Equivalence of Definitions of Limit of Real Function

Proof
By definition of deleted $\delta$-neighborhood of $c$:


 * $x \in \map {N_\delta} c \setminus \set c$


 * $0 < \size {x - c} < \delta$
 * $0 < \size {x - c} < \delta$

By definition of $\epsilon$-neighborhood of $L$:


 * $\map f x \in \map {N_\epsilon} L$


 * $\size {\map f x - L} < \epsilon$
 * $\size {\map f x - L} < \epsilon$

The result follows by comparison of the definitions.