Question about transaction fees

Firstly please let me know if this type of technical question should be posted somewhere else.
I wanted to know if I did not care about the speed of the transaction but did care that it does eventually go through. Does that mean I can use 1gwei? or will that just end up with a lost transaction?
And if not then would it no make sense to allow such transactions? meaning miners are already making profit from high transaction fees so they should oblige to ensure all transactions are eventually (within a reasonable period of time) processed? Please let me know if there is documentation or article that already addresses this if it already exists, thanks.