2.1 Basic Concepts: Convergence and Limits 53is infinitely large or infinitely small, we must find a way to express the concept
of infinity using only finite quantities. I call this the problem of "finitizing the
infinite." It was a problem of critical importance in the development of anal-
ysis as a rigorous subject, and was solved in the nineteenth century by Cauchy,
Weierstrass, and others. Their remarkable discovery was that inequalities and
quantifiers provide the perfect mathematical tools for "finitizing the infinite."
Coming into this course you may not feel very comfortable with either
inequalities or quantifiers. You haven't had to use them nearly as often as you
have used equations. That is why the mental change of gears is necessary. In
order to succeed in analysis, you will have to become quite skilled in handling
both inequalities and quantifiers.
Strategy for using Definition 2.1.4 to prove that lim Xn = L :
n-->oo- Start by letting E: denote an arbitrary positive real number. That means,
simply assume E: > O; you know nothing about E: other than it is positive. - Examine the inequality lxn - LI < E:. Try to find out how large n must be
in order to guarantee that lxn - LI < E:. This amounts to "playing with
inequalities." - Once you think you have found a value for no that will guarantee that
n ~ no => lxn - LI < E:, you must prove that this implication is true.
The following examples will illustrate important methods to be used in
implementing the definition of lim Xn = L. Pay careful attention to them.
n-->oo
fa~ [
Example 2.1.5 Consider. the limit.. statement n~. (2n + 3)^2
3
n _
7
= 3 ..( ) a Af ter h ow many terms are we guarantee d t h at ---2n + 3. is wit. h" m a d. istance
3n-7
of .01 of 2/3?(b) After how many terms are we guaranteed that the nth term of this se-
quence is an accurate approximation of the limit, to within 3 decimal
places?2n+3
(c) For arbitrary E: > 0, after how many terms are we guaranteed that -
3--
n - 7
is within a distance E: of 2/3?