# Thread:

1. ## Sequence and limit

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

2. Originally Posted by ashamrock415
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

$\{x_n\}$ is a monotone decreasing sequence, and as $c-\frac{1}{n}\geq c-1$ , we get that this sequence is bounded from below

and thus $\lim\limits_{n\to\infty}x_n$ exists.

All is left is to show that $c$ is actually this sequence's infimum...

Tonio
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