Single Variable Calculus, Chapter 2, 2.4, Section 2.4, Problem 24

Show that the statement $\displaystyle\lim\limits_{x \to a} c = c$ is correct using the $\varepsilon$, $\delta$ definition of limit.

Based from the defintion,


$
\begin{equation}
\begin{aligned}

\phantom{x} \text{if } & 0 < |x - a| < \delta
\qquad \text{ then } \qquad
|f(x) - L| < \varepsilon\\

\phantom{x} \text{if } & 0 < |x-a| < \delta
\qquad \text{ then } \qquad
|c-c| < \varepsilon\\

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}
& \text{That is,}\\
& \phantom{x} & \text{ if } 0 < |x-a| < \delta \qquad \text{ then } \qquad 0 < \varepsilon\\
\end{aligned}
\end{equation}
$

According to the definition...
$\quad \lim\limits_{x \to a} f(x) = L$

If for every number $\varepsilon > 0$ there is a number $\delta > 0$ such that
$\quad \text{if } \, 0 < |x-a| < \delta \quad \text{ then } \quad |f(x) - L| < \varepsilon$

Which means $\delta = \varepsilon$
Therefore,
$\quad \lim\limits_{x \to a}c=c $