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Closure Property of Natural Numbers

Ok so I have read the article and watched the shorthand proof on numberphile. However don't the natural numbers have a closure property, meaning if you add any of the natural numbers together you must get a natural number? So how can it be that when we add all of them, we get -1/12, which does not belong to the natural numbers? I think my idea of adding all the natural numbers doesn't match with what is going on in the article, so if anyone could fill in the gaps that would be great.

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