I see trees on Ancestry.com that indicate the 1880 US census provides proof of a birthdate in a specific location, say 4 April 1843 in Milroy, Rush County, Indiana. This confuses me just a little.
To get the definitions out of the way--genealogists usually don't refer to one document as "proof." Proof is the writing up of the information you have obtained through research and using that information and sound reasoning to reach a conclusion. The proof need not be overly verbose or extremely lengthy, but usually constructing a proof requires more than one piece of information.
We would say, usually, that the census provides evidence--or information. Documents and sources usually provide information and not proof.
However one needs to be careful with the 1880 census enumeration and carefully think about what evidence it provides and what evidence it does not. I would not say the 1880 census provides evidence of a birth in 4 April 1843 in Milroy, Rush County, Indiana. The 1880 census usually only provides places of birth as precise as the state and ages as precise as the year.
So the 1880 census for someone could easily indicate they were born in Indiana about 1843--the "about" because the year of birth depends upon the relationship between the date of birth and the date of the census. Our imaginary person could have had their birthday the day after the census taker came.
So if I used the Ancestry.com trees, should I "tie" an 1880 census record to a 4 April 1843 birth in Milroy, Rush County, Indiana? I don't think so. The reason is that is indicating the census indicates something that it does not. I should use the 1880 census as a source, but as a source of information less specific than the 4 April 1843 birth in Milroy.
A 37 year old native of Indiana in 1880 could easily have been born in Hendricks County or somewhere else within the state. And that 37 year old could easily have been born 31 May 1843 as well. That 1880 census enumeration provides evidence of a birth of 1842-1843 in Indiana--and nothing else. That is the birth information I should link to the 1880 census enumeration.
Of course this information is consistent with a 4 April 1843 birth in Milroy, Rush County, Indiana, but just because it is consistent with it does not eliminate other possibilities. It takes some concentration and attention to detail, but avoid indicating records say things that they do not.
As for "proof," I'll have to look at all the records I have on this person's date and place of birth and analyze them as a group.
After all, I want to be as correct as I can about the relative's date of birth.