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The argument to prove that there are different sizes of infinity is by saying .. no matter how many decimal numbers you write you can always come up with new one different from all others listed already.So, there is no one-one correspondence between natural numbers and decimal numbers. But, every time you come up with new one, you can add that new one to the list and correspond that to a natural number(which can go to infinity). So, again we have one-one correspondence and same cardinality and Hence same size of infinities. Please can someone help me what is wrong with my argument and what am I missing? It seems like the original argument is relying on the assumption that list of natural numbers end somewhere and we come up with a new decimal number to disprove one-one correspondence.