90 Mr Hardy, On the convergence 



It will be convenient to write 



The condition that a^n,n should be of bounded variation is then 

 that the series i | A„, a,„,,,i|, S|A„a^^,,j|, and iS ] A„i^,i«,„,„ j 

 should all be convergent. It is clear that these conditions in- 

 volve the regular convergence of a^^n to a limit a. 



Theorem 6. If the condition (2) is satisfied, and a„,,i and «!_„ 

 are of bounded variation in ni and n respectively, then a,n,n is 

 of bounded variation in (m, n). 



n — l 



For A^a^,„ = A^a^_i- 1, A^_^«.^,^, 



v = l 

 m—\ m— 1 ni-\n-\ 



S |A^a^,«!^ 2 |A^a^,i|+ S S |A^_^a^,J, 



/u. = l /u. = l n = l 1^ = 1 



so that 2 I A^,a^,M j 



is convergent. 



Theorem 7. If a^n,n is of bounded variation in {m, n), then 

 a,n= lim am.^n, ««= lim cv,n 



are of bounded variation in m and n respectively. 



For a^= ai 1,— % A 



^t=i 



/xfi^Ai, »»' 



W-1 71-1 ao 11-1 



S I «;, — a^+i| ^ 2 I A^tti,^ j + 2 2) I Aj^,^a^,„|, 



and so 2 ] a^ — a^+j j 



is convergent. 



Theorem 8. The necessary and sufiicient condition that a,n,n 

 should be of bounded variation is that its real and imaginary parts 

 should be of bounded variation. 



This follows from Theorem 1 and the inequalities 



I ^m,n(^'m,n \ ^ j ^m, ?i^m,9i I "r | ^in,n Hm.,n |j 

 where am,n = '^m„n + i^m,n- 



