A calculus exam answer sheet has the statement "A positive, monotone, decreasing sequence always converges to 0" marked as true, however, it seems demonstrably false. For instance, if we have the function (1/n) +1 define the values of a sub n, it seems to fit all the criteria: Positive from one to infinity, decreasing from 1 to infinity, and therefore monotone, but it converges to 1. Am I missing something, or is the answer sheet wrong?
The answer sheet is wrong.
Given any positive, monotonote, decreasing sequence that converges to $0$, we can always add a positive number $k$ to every term of the sequence to obtain a positive, monotone, decreasing sequence that converges to $k$.