1/n is basically "one divided into n parts"... using this definition, "one divided into an uncountable number of parts" will be naturally close to zero. I think the jump you made was
- infinity is uncountable
- 1/2, 1/3, 1/4.... 1/infinity
- 1/infinity is uncountable (Why? 1/infinity is clearly defined as zero..but let's say I grant you this even if I do not agree)
- infinity is less than zero (Why? just because it is uncountable, it is not infinite )
Cheers!