By Kenneth Hoffman

Constructed for an introductory path in mathematical research at MIT, this article makes a speciality of techniques, rules, and strategies. The introductions to actual and complicated research are heavily formulated, they usually represent a traditional advent to complicated functionality concept. Supplementary fabric and routines look through the textual content. 1975 version.

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An_ 1 have been determined, an is the largest integer k such that a110-1 + a210-2 + ... + an-1 110-cn- 11 + k 10-n < X. n- 10-(n- What we have done is to place x successively in the semi-closed intervals J1, J2, J3, ... 7) in = [a110-1 + a210-2 + ... + an10-n, a110-1 + a210-2 + . . +. (an + 1)10- n). The intervals in are "nested" J1 :-) J2 :-) J3 and x belongs to the intersection of all the Jn. 8) that is, no other number belongs to every Jn. Why? If a, b E in, then a - b I < l0-n. If y (as well as x) belongs to everyJn, then Iy - xI< 10-n, n= 1,2,3,....

EXAMPLE 4. One of the most useful special cases of Theorem 3 is the following. Suppose [Xn} is a sequence and Convergence Criteria Sec. 13) n = 1,2,3,... IXn-Xn+1I <2-n, Then [Xn} is a Cauchy sequence. Why? Suppose k < n. Then I Xk Xn I f I Xk Xk+ 1 I + I Xk+ 1 Xk+2 I + * * '± I Xn- 1 Xn I < 2-k +2- (k+ 1) + . . +2- (n-1) = 2(2-k - 2-n) < 2-(k-1). 13) is convergent. EXAMPLE 5. The Cauchy condition on a particular sequence frequently arises in this way. In addition to the sequence of points Xn, we have a sequence of sets S1, S2, S3, ...

XnEn. Theorem 6. If S is a subspace of Rn, then S has a basis, and every basis for S consists of precisely dim S vectors. Proof. If d = dim S, then S has a basis consisting of d vectors: We can find vectors Y1, ... , Yd in S which are independent. By the definition of d, we know that, for any X EE S, the vectors Y1, ... , Yd, X are dependent. But that means that X is a linear combination of the vectors Y;. Suppose V1, ... , Vk is any (ordered) basis for the subspace S. Each X in S can be expressed as a linear combination X = Cl V1 + .

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