# Do positive, monotone, decreasing sequences always converge at 0?

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, as the idea of your example shows. But you haven't defined a sequence; $a_n = 1 + 1/n$ is a sequence, but your thing doesn't have an $n$ in it (even though you called it $a_n$...). – user296602 Apr 18 '18 at 1:22
• Any monotone decreasing sequence $\{a_n\}$ the converges to $c$ can be shifted to have and another monotone decreasing sequence $\{b_n\} = \{a_n - c + M\}$ that can converge to any value of $M$. And if $M \ge 0$ then $\{b_n\}$ is positive. The answer sheet is not just wrong. It's incomprehensible and inconsistantly and logically incompatibly wrong. – fleablood Apr 18 '18 at 1:35

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 $k$.