By C. H. Edwards Jr.

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**Extra info for Advanced Calculus of Several Variables**

**Example text**

In order to prove that lim x ^ (1 n / ( x , y) = 3, we first write \f(X9y)-3\ = \χ2 +Xy+y-3\ 2 ^ \x - l| + \y\f(x,y)-3\ 1[ + \xy; l| = | χ + ΐ| \χ-ι\ + ( j - i | + \xy-y Z\x+l\\x-\\ + 2 | ^ - l | + b | \x-\\. +y - i \ , (I) Given ε > 0, we want to find δ > 0 such that |(x, y) - (l, l)| = [(x - l) 2 + (y — l) 2 ] 1 / 2 < δ implies that the right-hand side of (I) is <ε. Clearly we need 7 Limits and Continuity 43 bounds for the coefficients |x + 11 and |>'| of \x - 11 in (1). So let us first agree to choose δ ^ 1, so | ( x , j O - ( l , 1)[ < ( 5 = > | x - 1| < 1 and =>0

Yn) e @m+n. If A c ^ m and 5 c @n9 then the Cartesian product A x B = {(a, b)e@m+n:*eA is a subset of @ and b e B} m+n . 5 If A is a compact subset of Mm and i? is a compact subset of 9tn9 then ,4 x £ is a compact subset of @m+n. PROOF Given a sequence {c,}f ={(a f , b,)}f of points of A x B, we want to show that it has a subsequence converging to a point of A x /?. ^ has a subsequence {a^JLi which converges to a point Sie A. }*= i has a subsequence {b(/;k)}^= { which converges to a point be B.

Ii') The intersection of any collection of closed sets is a closed set. (iii') The union of a finite number of closed sets is a closed set. Examples: (a) A closed interval is a closed subset of 01. 3). (c) A finite set F of points is a closed set. (d) The real line 01 is a closed subset of 0l2. ) The last example illustrates the following useful alternative characterization of closed sets. 1 The subset A of0ln is closed if and only if it contains all of its limit points. PROOF Suppose A is closed, and that a is a limit point of A.