142 



Algebra. 



We are to prove that a^b. If a and b are not equal, suppose a 



greater than b and let 



a — b=d 



Let a—x=u and b—y^=i\ 



then a—x-^2i and b=y-\-v, 



and a — b=d becomes by substitution, 



(x + u)-(y + v) = d 

 or (x—y)-\-(u—v) = d 



Since lim x==a, lim u=o and as lim _r=<^, lim i'=o, or z/ and z' 

 are each variables which can be made as small as we please, and 

 hence the difference u — v can be made as small as we please, and 

 so can be made so small as not to cancel d, hence x—y would 

 equal something, or x and jk would differ, which is contrary to the 

 hypothesis; hence a cannot be greater than b, and in the same 

 way it may be shown that b cannot be greater than a. 



Therefore a=b. 



8. Theorem. The limit of the algebraic sum of several variables 

 equals the algebraic siun of their separate limits. 



IvCt the variables be x,y, r, etc., and let lim .1 = ^', lim i'=^, 

 lim ,2-=^, etc., we are to prove 



limr-r+J+2-+ . . . ) = (a^b^c-\- . . . ) 



Ivet 



then x-hr + ,2'4- . . . =(a^b-^c^ . . )-(u^-v-V7v-\- . . ;. 



Suppose // to be numerically the greatest of the quantities 

 II, V, IV, . . . and suppose that there are 7i of these quantities. 



Now, since x may be taken so near a as to differ from it b}- an 



amount as small as we please, we may take x so that 



k 

 u<i , 

 11 



or nu<^k, 



however small k may be. 



Then u -\- v -\- zv -\- . . . <;^^/ (since u is the largest of the 



quantities ?/, v, w, . . . ). Hence u -\- v -\- 7u -\- . . . </^, however 



small k may be ; that is, x-\-y-\-z-\- . . . may be made to differ 



fram a-\-b-\-c . . . by an amount as small as we please. Hence 



limit (x+y+.z+ . . . ) = a + b^c-\- . . . 



