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?