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By V Sankrithi Krishnan

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**Example text**

B. for a definite value of 2. This is illustrated by the following example. f(s)= - 3~ f 4Czs+ 65' - 1 2 ~ 5 , where c is a real number for which it is unknown whether c>O, c = O or c t O . f'(s) = - 12(5+1) (5- 1 ) ( 5 - c ) . f ( - 1) = 3 + 8c, f ( 1)= 3 - 8c, f ( c ) =c4- 6~'. The least upper bound of f ( z ) is 3 + 8 IcI, but it is unknown whether f ( z ) takes this value for z = - 1 or for x = 1. 6. The Bolzano- Weierstras theorem Brouwer investigated this theorem (L. E. J. Brouwer 1962Bl.

D e f i n i t i o n 4 . Two pg-species are geometrically congruent if neither of them can contain a member that cannot coincide with a member of the other. Here also analogous definitions must be given, as in the case of def. 3. 3. Distance and topology D e f i n i t i o n 1. h? Y2) is max (lzl-Yll, Iz2-Y81). Remark. The theory could be developed a13 well with VUX1- z2Y + (Yl-Y2I21 aa the diatance of x and y. The above definition is chosen for the sake of simplicity of formulas. s-neighbourhood and of a neighbourhood of a point p can be introduced in the usual way by means of this notion of distance.

If the vectors a,, . . , a, are free and the vectors 67 LINEAB DEPENDENOE 4,.. ,,,b are free, then at least one vector b, is free from +, . ,a,, b, are h). %, . . , a , (that is, Proof. The matrix of the a, has rank r ; we may suppose that the determinant d formed out of its first r columns is # 0. ,Y ; a= 1, . , r 1) , (8= 1,. + r + 1) As the b, are free, there is a determinant, formed out of their components with subscripts jl, .. ,it+,, which is # 0. The corresponding determinant formed out of the c, is 0, so we find at leaat one pair of subscripts t , u with b , # c,.