By Maurice Heins

ISBN-10: 0080873405

ISBN-13: 9780080873404

ISBN-10: 0123379504

ISBN-13: 9780123379504

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**Example text**

Uniformly) to f I Y. ) tends tofpointwise. Show by example that the converse is false. 5. Suppose thatf, tends uniformly tofand g . tends uniformly tog, g and g , mapping X into C. Show that (f. ) need not tend uniformly to a complex-valued function with domain X. It is to be observed that the following definitions are in force. Let A denote a ring, let f, g : X +A . Then byff g is meant the map taking x intof(x) g(x) and by f g is meant the map taking x into f ( x ) g ( x ) . With addition and multiplication so defined the set off: X +.

The notion of a limit may be viewed in the following manner. Let 2 denote a ) where (a,,) is a real sequence, a E R, and the set of ordered pairs ((a,JnPN, (a,,) has limit a. It might be well to note that implicit in taking for granted that “ lim a,, ” is meaningful is the assumption that (a,) belongs to the domain of 2,that is, that for some a E R the sequence (a,,) has limit a. These remarks hold elsewhere in the study of limits. Corresponding observations hold for limit superior and limit inferior ($6, Ch.

Invoking (C) for q = 1 we see that (a,,) is bounded. Let u p = sup{a,+,,: n E N } and let p, = inf{u,+,: n E N}. Given positive q we see that from a,+, - aq < q, p , q 2 m and n E N, we have a, - a4I q, p , q 2 m,and hence up - p, I q for p 2 m. Hence lim ap = lim p,. We denote the common value by a. We assert that lim a, = a. Indeed, given positive q, for some p we have + q, a- q < p, I a, < u u-q < a,, < u + q, n 2 p . and hence BB Chapter I.. @@ T H E COMPLEX F I E L D . L I M I T S PREFATORY REMARKS It is a commonplace that R is not aIgebraicaIIy closed; that is, it is not true that every polynomial of positive degree with real coefficients takes the value 0 at some x E R.