~The GLB is short for "the greatest lower bound" or "limit inferior" as some call it.

~The LUB is short for "the least upper bound" or "limit superior" as some call it.

~To understand these, we need to discuss the meaning of a "bound"

~A set is said to be bounded above, if there exists a number M, such that every member of the set is less than or equal to M.

~For example, the set  {-1, 0, 3, 7}  is bounded above by many numbers, so there are many M's that satisfy the definition. The number 20 is an upper bound since all members of the set are less than or equal to 20. Matter of fact, any number 7 or greater could be used.

~7 is a special one, however. It is the smallest. This is the LUB (the least upper bound)

~Similarly, a set is said to be bounded below, if there exists a number m, such that, all members of the set are greater than or equal to m. In the example given, any number less than or equal to -1 would serve as an m.

~-1 is a special one, however. It is the largest. This is the GLB (the greatest lower bound)

~If our set consists of y values on the graph of some curve, the concepts are the same, however, you, most likely, will be dealing with an infinite set.

~For example, take the curve f(x)=x² over (0;3).  Note that this is an open interval and 0 and 3 are not included. (there are no maximum or minimum values)(see link on Max-Min in the review section)
The graph consists of just a piece of the basic parabola f(x)=x².

~This function is certainly bounded above by many values of M. (9 or greater) and bounded below by many values of m (0 or less).

~We say that the curve is bounded, if it is both bounded above and below. So, this curve is a bounded curve.

~Graphically, one can visualize this by taking note that the curve can lie intirely between two horizontal lines.

~The LUB{f(x)} = 9  (which simply says that the least upper bound on the set of y values is 9)

~The GLB{f(x)} = 0 (which simply says that the greatest lower bound on the set of y values is 0)

~If the y values of a curve increase (or possibly stay the same) as we move from left to right on the x axis, we say that it's a monotonic or monotone function. A monotonic function whose values are always increasing, is said to be strictly increasing.

~Similarly, if the y values of a curve decrease ( or possibly stay the same) as we move from left to right on the x axis, we also say it's monotonic or monotone. To distinguish between going up or going down, we add the words increasing or decreasing.

~So, if someone tells you a function is monotonic, you really don't know .But if they say, monotonic increasing or monotonic decreasing, that will make it clear. The word strictly is used to eliminate the possibility that the y values are equal.

~unfortunately, not all math teachers use the same "lingo" when explaining these concepts.

~An important feature combining the concepts of boundedness and a monotonic characteristic is the following:

   If a function is bounded and is monotonic, then it has a limit as x    
   approaches infinity.  Also, that limit will be the LUB or the GLB.
   Think about it.

~An important place in calculus where this is useful concerns the existence of area under a graph between 2 x values. In more theorically based courses, lower and upper sums are used. Areas of outer and inner polygons that trap the region to be defined as the area can be used. Usually, we use polygons formed by sums of circumscribed (upper sums) and inscribed rectangles (lower sums) so that specific fomulation can be used relative to the curve in question. By letting the number of rectanges increase to infinity, the area can be defined as the LUB or the GLB of those sums, if they converge to one value.(i.e., LUB{LOWER SUMS}=GLB{UPPER SUMS}=THE AREA). If these do not converge to the same value, then the area does not exist.

~Not all bounded functions have limits as x approaches infinity. Take, for example, f(x) = sin(x). It is certainly bounded. The LUB=1 while the
GLB= -1.  However, as x approaches infinity, the curve oscillates between 1 and -1, hence no limit.