Home » Uncategorized » “Upper” bound

# “Upper” bound

Let be a function satisfying

for all positive real numbers and . Prove that

Solution

For starters, let us assume that . Dividing the interval into subintervals each of length so that . Thus,

The inequality implies that

Hence,

The limit exists and equals to . Hence , the inequality is proved for .

Now, assume that . Dividing the interval into subintervals each of length so that . Thus,

The inequality implies that

Hence,

The limit exists and equals to . Hence , the inequality is also proved for . This completes the proof!