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Proof

(related to Proposition: Limit of the Identity Function)

Context

• Let $a\in\mathbb R.$
• Let $f:\mathbb R\to\mathbb R,$ $f(x)=x$ be the identity function.

Hypothesis

• Given $\epsilon > 0$.

Implications

• Then for $\delta:=\epsilon$ it holds that for all $x$ with $0 < |x-a| < \delta$ we have that $|f(x)-a|=|x-a| < \epsilon.$

Conclusion

• By the definition of limit, we have $\lim_{x\to a }f(x)=a.$
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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016