Enough proof? ?

no, but asking this multiple times on here would be...i suspect you don't have a clue what you are talking about and are actually liable..likely signed something WITH THE LANDLORD you didn't read and just don't remember.

the council has nothing to do with what you signed with the landlord.....

You lived there, you were a tenant.  

You keep asking the same questions.....the council have more to do than writing you letters especially as they can't say if what you say is true or not.............. FACT is the landlord WON the case against you and your mother and would have had to prove in court papers that you were both tenants and neither of you turned up in court ..........

You back again?  We have no idea how the court will treat a letter from the council but if not adding you saved your mother money (for example if she claimed the single person discount for council tax), the judge may decide that your mother lied to get extra benefits.  You need to provide documents SHOWING THAT YOU LIVED ELSEWHERE during the time in question.  If you cannot, you will not win.

No because you DID live there for 1 month. You WERE a tenant. You your self say you were a nonliable TENANT. 

I have no idea what you're trying to prove.  But if you lived there for a month, you were a tenant for the month.