# Thread: Proof of Lemma on Polynomials

1. ## Proof of Lemma on Polynomials

There is a Lemma in my textbook that states,

If P(h) is a polynomial of degree <= k, that vanishes to order > k as h -> 0 [i.e. P(h)/|h|^k -> 0], then P === 0 (=== is a triple equals sign)

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I don't quite understand what P===0 means. I know that the === represents congruence used in modular arithmetic. However, how is this applicable here?

Also, I don't understand how P(h)/|h|^k -> 0 implies that polynomial of degree <= k vanishes to order > k. From this statement, it seems like P vanishes to order equal to k.

Basically, what does this Lemma even mean?

2. If $P \in F[x]$ (where $F= \mathbb{R}$ or $\mathbb{C}$ ) $P(x)= \sum_{j=0} ^{k} \ a_jx^j$ and $\frac{P(x)}{x^k} = \sum_{j=0} ^{k} \ a_jx^{j-k}$ and so for every $i the term $x^{i-k} \rightarrow \infty$ as $x \rightarrow 0$ (assuming $a_i \neq 0$) and so for all $i $a_i=0$, and so $0= \frac{P(x)}{x^k} = a_k$ and so $P \equiv 0$ (this means P is identically zero). Now try induction on the number of variables.