For all values of n ii if is any positive number then

Info icon This preview shows pages 196–199. Sign up to view the full content.

for all values of n ; (ii) if is any positive number then φ ( n ) > M - for at least one value of n . This number M we call the upper bound of φ ( n ). Similarly, if φ ( n ) is bounded below, that is to say if there is a number k such that φ ( n ) 5 k for all values of n , then there is a number m such that (i) φ ( n ) = m for all values of n ; (ii) if is any positive number then φ ( n ) < m + for at least one value of n . This number m we call the lower bound of φ ( n ). If K exists, M 5 K ; if k exists, m = k ; and if both k and K exist then k 5 m 5 M 5 K. 82. The limits of indetermination of a bounded function. Sup- pose that φ ( n ) is a bounded function, and M and m its upper and lower bounds. Let us take any real number ξ , and consider now the relations of inequality which
Image of page 196

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

[IV : 82] POSITIVE INTEGRAL VARIABLE 181 may hold between ξ and the values assumed by φ ( n ) for large values of n . There are three mutually exclusive possibilities: (1) ξ = φ ( n ) for all sufficiently large values of n ; (2) ξ 5 φ ( n ) for all sufficiently large values of n ; (3) ξ < φ ( n ) for an infinity of values of n , and also ξ > φ ( n ) for an infinity of values of n . In case (1) we shall say that ξ is a superior number, in case (2) that it is an inferior number, and in case (3) that it is an intermediate number. It is plain that no superior number can be less than m , and no inferior number greater than M . Let us consider the aggregate of all superior numbers. It is bounded below, since none of its members are less than m , and has therefore a lower bound, which we shall denote by Λ. Similarly the aggregate of inferior numbers has an upper bound, which we denote by λ . We call Λ and λ respectively the upper and lower limits of indetermination of φ ( n ) as n tends to infinity ; and write Λ = lim φ ( n ) , λ = lim φ ( n ) . These numbers have the following properties: (1) m 5 λ 5 Λ 5 M ; (2) Λ and λ are the upper and lower bounds of the aggregate of intermediate numbers, if any such exist; (3) if is any positive number, then φ ( n ) < Λ + for all sufficiently large values of n , and φ ( n ) > Λ - for an infinity of values of n ; (4) similarly φ ( n ) > λ - for all sufficiently large values of n , and φ ( n ) < λ + for an infinity of values of n ; (5) the necessary and sufficient condition that φ ( n ) should tend to a limit is that Λ = λ , and in this case the limit is l , the common value of λ and Λ. Of these properties, (1) is an immediate consequence of the definitions; and we can prove (2) as follows. If Λ = λ = l , there can be at most one intermediate number, viz. l , and there is nothing to prove. Suppose then that Λ > λ . Any intermediate number ξ is less than any superior and greater than any inferior number, so that λ 5 ξ 5 Λ. But if λ < ξ < Λ then ξ must be intermediate, since it is plainly neither superior nor inferior. Hence there are intermediate numbers as near as we please to either λ or Λ.
Image of page 197
[IV : 82] LIMITS OF FUNCTIONS OF A 182 To prove (3) we observe that Λ + is superior and Λ - intermediate or inferior. The result is then an immediate consequence of the definitions; and the proof of (4) is substantially the same.
Image of page 198

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 199
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern