Let xn be a sequence of real numbers satisfying xn+1≤xn for n=1,2,...
Assume there is a constant c such that xn>c-1/n.
Show that l=lim(xn) exists and l>c
$\displaystyle \{x_n\}$ is a monotone decreasing sequence, and as $\displaystyle c-\frac{1}{n}\geq c-1$ , we get that this sequence is bounded from below
and thus $\displaystyle \lim\limits_{n\to\infty}x_n$ exists.
All is left is to show that $\displaystyle c$ is actually this sequence's infimum...
Tonio