Is infinity an element of the natural numbers?

No, it is not. ${}{}{}{}{}{}{}{}$

The natural numbers are, almost by definition:

• 0 (sometimes)
• the number after 0
• the number after that
• the number after that
• the number after that
• etc.

Nowhere in that list is $\infty$.

(Note: Sometimes, 0 isn't considered a natural number. Then just replace 0 by 1.)

EDIT: I agree with many of the comments. I guess I am being informal with my language here, and that example was not so great. It is now too late to delete this answer, but consider it deleted.