Conference rusure::math

793.0. "question on metric space l^2" by SMURF::MCMENEMY (Michael G. McMenemy) Sun Nov 29 1987 17:16

In reviewing, this following proof a question arose?

---------------------------------------------------------------
Definition:

         2
	l  denotes the set of all real sequences { a } for which the series
                                                    k
            oo
	  -----	  2
	  \	 a
	  / 	  k       converges.
          ----
           k=1


Corollary:
                                      2
	If { a , b } are elements of l, then the series
              k   k
        
            oo
	  -----	         2
	  \	( a - b )
	  / 	   k   k    converges.
          ----
           k=1


Proof:

	

The series  
	    oo
	  -----	         2
	  \	( a - b )
	  / 	   k   k    is the sum of the three convergent series
          -----
           k=1

            oo
	  -----	  2
	  \	 a 
	  / 	   k  ,
          -----
           k=1

            oo
	  -----	         
     - 2  \	 a * b 
      	  / 	  k   k  , 
          ----
           k=1
 
           oo
	  -----   2
	  \	 b 
	  / 	  k
          ----
           k=1


	and therefore is converegent by the fact that if two or more
	series converge then the sum of these series also converges.


	Finally, here comes the question. How do you show (prove) that
	the following series converges?

            oo
	  -----	         
     - 2  \	 a * b 
      	  / 	  k   k  , 
          ----
           k=1



				Thanks Mike.

                        2   2      2   2
Since |a * b | <= max (a , b ) <= a + b ,
        k   k           k   k      k   k

      ---               ---               ---      2   2       ---  2   2
abs ( >   a * b  )  <=  >   |a * b |  <=  >   max(a , b )  <=  >   a + b
      ---  k   k        ---   k   k       ---      k   k       ---  k   k