Equivalence of Definitions of Limit of Real Function

Theorem

The following definitions of the concept of Limit of Real Function are equivalent:

Definition 1

$\map f x$ tends to the limit $L$ as $x$ tends to $c$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$

where $\R_{>0}$ denotes the set of strictly positive real numbers.

Definition 2

$\map f x$ tends to the limit $L$ as $x$ tends to $c$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: x \in \map {N_\delta} c \setminus \set c \implies \map f x \in \map {N_\epsilon} L$

where:

$\map {N_\epsilon} L$ denotes the $\epsilon$-neighborhood of $L$

$\map {N_\delta} c \setminus \set c$ denotes the deleted $\delta$-neighborhood of $c$

$\R_{>0}$ denotes the set of strictly positive real numbers.

Proof

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

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

if and only if:

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

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

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

if and only if:

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

The result follows by comparison of the definitions.

$\blacksquare$