Number Theory: An Introduction to Mathematics

3 Completions 269

It will now be shown that the procedure by which Cantor extended the field of
rational numbers to the field of real numbers can be generalized to any valued field.
LetFbe a field with an absolute value||. A sequence (an)ofelementsofFis said
toconvergeto an elementaofF,andais said to be thelimitof the sequence (an), if
for each realε>0 there is a corresponding positive integerN=N(ε)such that

|an−a|<ε for alln≥N.

It is easily seen that the limit of a convergent sequence is uniquely determined.
A sequence(an)of elements ofFis said to be afundamental sequenceif for each
ε>0 there is a corresponding positive integerN=N(ε)such that

|am−an|<ε for allm,n≥N.

Any convergent sequence is a fundamental sequence, since

|am−an|≤|am−a|+|an−a|,

but the converse need not hold. However, any fundamental sequence is bounded since,
ifm=N( 1 ), then forn≥mwe have

|an|≤|am−an|+|am|< 1 +|am|.

Thus|an|≤μfor alln,whereμ=max{|a 1 |,...,|am− 1 |, 1 +|am|}.
The preceding definitions are specializations of the definitions for an arbitrary met-
ric space (cf. Chapter I,§4). We now take advantage of the algebraic structure ofF.Let
A=(an)andB=(bn)be two fundamental sequences. We writeA=Bifan=bn
for alln, and we define the sum and product ofAandBto be the sequences

A+B=(an+bn), AB=(anbn).

These are again fundamental sequences. For we can chooseμ≥1sothat|an|≤μ,
|bn|≤μfor allnand then choose a positive integerNso that

|am−an|<ε/ 2 μ, |bm−bn|<ε/ 2 μ for allm,n≥N.

It follows that, for allm,n≥N,

|(am+bm)−(an+bn)|≤|am−an|+|bm−bn|<ε/ 2 μ+ε/ 2 μ≤ε,

and similarly

|ambm−anbn|≤|am−an||bm|+|an||bm−bn|<(ε/ 2 μ)μ+(ε/ 2 μ)μ=ε.

It is easily seen that the setFof all fundamental sequences is a commutative ring
with respect to these operations. The subset of all constant sequences(a),i.e.an=a
for alln, forms a field isomorphic toF. Thus we may regardFas embedded inF.
LetN denote the subset ofFconsisting of all sequences(an)which converge
to 0. EvidentlyN is a subring ofFand actually an ideal, since any fundamental
sequence is bounded. We will show thatN is even a maximal ideal.