Proof: By Induction
(related to Proposition: Limit of a Polynomial)
Context
- Let $n$ be a natural number.
- Let $a, a_n,a_{n-1},\ldots,a_1,a_0\in\mathbb R$ be real numbers.
- Let $p:\mathbb R\to\mathbb R,$ $p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ be a polynomial of the degree $n.$
- We will provide a proof for the limit.
$$\lim_{x\to a }p(x)=p(a)$$
on the degree $n.$
Base Case
- For $n=0$ we have that $p(x)=a_0$.
- By the limit of the constant function it follows that $\lim_{x\to a}a_0=a_0.$
Hypothesis
- Assume that for some natural number $k$, if $p(x)=a_kx^k+a_{k-1}x^{k-1}+\ldots+a_1x+a_0$, then $\lim_{x\to a }p(x)=p(a)=a_ka^k+a_{k-1}a^{k-1}+\ldots+a_1a+a_0$.
Induction Step
- If if $p(x)=a_{k+1}x^{k+1}+a_kx^k+a_{k-1}x^{k-1}+\ldots+a_1x+a_0,$ then by the limit of $n$-th powers and the limit of product it follows $$\begin{array}{rcl}
\lim_{x\to a }p(x)&=&\lim_{x\to a }a_{k+1}x^{k+1}+a_kx^k+a_{k-1}x^{k-1}+\ldots+a_1x+a_0\\
&=&(\lim_{x\to a }a_{k+1}x^{k+1})+(\lim_{x\to a }a_kx^k+a_{k-1}x^{k-1}+\ldots+a_1x+a_0)\\
&=&(\lim_{x\to a }a_{k+1})(\lim_{x\to a }x^{k+1})+a_ka^k+a_{k-1}a^{k-1}+\ldots+a_1a+a_0\\
&=&a_{k+1}a^{k+1}+a_ka^k+a_{k-1}a^{k-1}+\ldots+a_1a+a_0\\
&=&p(a).
\end{array}$$
Conclusion
- By induction, it follows that $\lim_{x\to a }p(x)=p(a)$ is true for all $n\ge 0.$
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References
Bibliography
- Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016