J/x, y, zG Rm and A €E M then (i) x • x > 0 with equality if and only г/ х = 0, (ii) x • у = у • x,
(Hi) (Ax) -y = A(x-y),
(iv) (x + y) • z = x • z + у • z.

Although the proof just given is fairly detailed, it is a worthwhile exercise to extend it so that all the steps are directly justified by reference to the properties of the inner product given in Lemma 4-1-

J/x = (x\, X2, xm) 6 Rm show that
m
i i и и \—^ II II
max \хЛ < x < > \хЛ < m max \хЛ.

We work in Rm with the Euclidean norm. We say that a sequence ai, аг, ... tends to a limit a as n tends to infinity, or more briefly
an —»• a as n —»• oo,
if, given e > 0, we can find an no (б) such that
||an — a|| < 6 for all n > no(e).

IRm and there exists a К such that ||xn|| < К for all n, then we can find n(l) < n(2) < ... and x 6 W11 such that ^nu) —> x as j —»• oo. Once again 'any bounded sequence has a convergent subsequence'.

When we work in R the intervals are, in some sense, the 'natural' sets to consider. One of the problems that we face when we try to do analysis in many dimensions is that the types of sets with which we have to deal are much more diverse.

A set F С ]Rm is closed if whenever xn 6 F for each n
and xn —»• x as n —»• oo then x 6 F.
Thus a set is closed in the sense of analysis if it is 'closed under the operation of taking limits'. An indication of why this is good definition is given by the following version of the Bolzano-Weierstrass theorem.

(i) If A is a non-empty closed subset of Ж with supremum a, then we can find an 6 A with an —> a as n —> oo.
(ii) If A is a non-empty closed subset of Ж, then, if supa€yl a exists, supa€j4 aeA.
We turn now to the definition of an open set.

Consider sets in Rm. Let x 6 W71 and r > 0. (i) The set -B(x, r) = {y £ Rm : ||x — у || < r} is open. (ii) The set -B(x, r) = {y £ Rm : ||x — y|| < r} is closed.

A subset U of Rm is open if and only if each point of U is the centre of an open ball lying entirely within U.
Thus every point of an open set is surrounded by a ball consisting only of points of the set.
The topics of this section are often treated using the idea of neighbourhoods. We shall not use neighbourhoods very much but they come in useful from time to time.