Is the statement of the problem correct? The inequality holds trivialy without assuming any special relation between the terms of the sequence except the assumption concerning the positivity of the sequence. This follows from the following observation:
.
Note that the left-hand side is greater than or equal to 1.
of positive real numbers such that for all 
is 
. For all natural 
prove the inequality
except the assumption concerning the positivity of the sequence. This follows from the following observation:
.