# 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

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