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Proof: By Induction

(related to Proposition: Limit of Nth Powers)

Context

• Let $n\ge 1$ be a natural number.
• Let $f:\mathbb R\to\mathbb R,$ $f(x)=x^n$ be a $n$-th power function.
• Let $a\in\mathbb R$ be a real number.
• It follows a proof by induction on $n$ for the limit $$\lim_{x\to a }f(x)=\lim_{x\to a }x^n=a^n.$$

Base Case

• When $n=1$, the statement says that $\lim_{x\to a}x^1=\lim_{x\to a} x = a$. This follows from the limit of the identify function.

Hypothesis

• Assume that for some natural number $k$, $\lim_{x\to a}x^k=a^k$ is true.

Induction Step

• According to the proposition of limit of the product, it follows that $$\lim_{x\to a}x^{k+1}=\lim_{x\to a}x^{k}\cdot x=(\lim_{x\to a}x^k)\cdot (\lim_{x\to a}x)=a^k\cdot a=a^{k+1}.$$
• Thus, the statement ist true for $n=k+1.$

Conclusion

• By induction, $$\lim_{x\to a }f(x)=\lim_{x\to a }x^n=a^n$$ is true for all $n\ge 1.$
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References

Bibliography

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