Give a sequence satisfying given condition: 1) 2) I have tried many many options but can\'t seem to get these two last ones quite right. Please show all details. Thanks Solution 1) the first one is impossible since if m is negative 1/m<0 and |a_n-m|>=0, so |a_n-m|<1/m is impossible If it is meant for every m in N , the natural numbers and we exclude 0, then you may choose a_n also the sequence of natural numbers, then for any m in N, you choose a_n=m, so |a_n- m|=0<1/m 2) choose a_n=(-1)^n, that is 1,-1,1,-1,... then the sequence of even partial sums is 1-1,1-1+1-1,..., that is constant 0,0,... which converges to 0.

1. Give a sequence satisfying given condition:
1)
2)
I have tried many many options but can't seem to get these two last ones quite right. Please show
all details.
Thanks
Solution
1) the first one is impossible since if m is negative
1/m<0 and |a_n-m|>=0, so |a_n-m|<1/m is impossible
If it is meant for every m in N , the natural numbers and we exclude 0, then you may choose a_n
also the sequence of natural numbers, then for any m in N, you choose a_n=m, so |a_n-
m|=0<1/m
2) choose a_n=(-1)^n, that is
1,-1,1,-1,...
then the sequence of even partial sums is 1-1,1-1+1-1,..., that is constant 0,0,... which converges
to 0