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Why the definition of limit is often written in this form also it can be more easy ?

in Real numbers and Real Analysis by Ethan.D.Bloch : writes the definition :

let I[itex]\subseteq [/itex]ℝ be an open interval ,c [itex]\in[/itex]I , let f:I-{c} → ℝ be a function and let L[itex]\in[/itex]ℝ , L is the limit of f as x goes to c ,

if for any ε>0 , there exists δ>0 such that

some questions concerned here , why he don't write instead of the

0<|x-c| < δ imply |f(x)-L| < ε

in the first definition does the inequality mean that there is some x satisfy it such that x [itex]\notin[/itex] I ?

in Real numbers and Real Analysis by Ethan.D.Bloch : writes the definition :

let I[itex]\subseteq [/itex]ℝ be an open interval ,c [itex]\in[/itex]I , let f:I-{c} → ℝ be a function and let L[itex]\in[/itex]ℝ , L is the limit of f as x goes to c ,

if for any ε>0 , there exists δ>0 such that

**x [itex]\in[/itex] I-{c} and |x-c| < δ imply |f(x)-L| < ε**some questions concerned here , why he don't write instead of the

**Bold**part this simply0<|x-c| < δ imply |f(x)-L| < ε

in the first definition does the inequality mean that there is some x satisfy it such that x [itex]\notin[/itex] I ?

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