1549901369-Elements_of_Real_Analysis__Denlinger_

2.9 *Upper and Lower Limits 131

is called the degree of x. For example, J3 is algebraic of degree 2, since

it satisfies x^2 - 3 = 0.

(a) Prove that Vn EN, there are only countably many algebraic numbers

of degree n.

(b) Prove that the set of algebraic numbers is countable.

( c) Prove the existence of transcendental numbers by proving that the

set of transcendental numbers is uncountable. [Recall how we proved

that the set of irrational numbers is uncountable.]

(d) Search the literature and find proofs that 7r and e are transcendental.

I suggest Niven, [100].

2.9 *Upper and Lower Limits

Not every sequence has a limit. But, after we make the appropriate definitions,
we shall see that every sequence has a unique "upper limit" and a unique "lower
limit,'' which may be a finite real number, or +oo or -oo. These upper and
lower limits will prove useful in Chapters 8 and 9, but will not really be needed
before then.

Definition 2.9.1 (Upper Limit) Suppose {xn} is any sequence ofreal num-
bers.

Case 1: If {xn} is bounded above we define, Vn EN,

Xn = sup{xk: k ~ n}.

That is, Xn is the supremum of the n-tail^18 of {xn}· Since {xn} is bounded

above, each Xn is a real number. Note that

We define the upper limit of { Xn} to be

n-+oo lim Xn = n-+oo lim Xn·

Since { xn} is monotone decreasing, we know that lim Xn is either a real

n-+=

number or -oo.

Case 2: If {xn} is not bounded above then Vn EN, Xn = +oo and we define

n-+oo lim Xn = n-+oo lim ( +oo) = +oo.

18. See Definition 2.2.15.