◀ ▲ ▶Branches / Analysis / Proof

Proof

(related to Proposition: Limit of the Constant Function)

Context

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

Hypothesis

• Given $\epsilon > 0$.

Implications

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

Conclusion

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

Thank you to the contributors under CC BY-SA 4.0!

Github:

References

Bibliography

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