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
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
is a monotone decreasing sequence, and as , we get that this sequence is bounded from below
and thus exists.
All is left is to show that is actually this sequence's infimum...