- #1
y_lindsay
- 17
- 0
if the real sequence {[tex]x_n[/tex]} satisfies that for every m>n, [tex]|x_n-x_m|>\frac{1}{n}[/tex], can we prove it's unbounded?
here is what i thought:
let's suppose the sequence is bounded, then there is its sub-sequence who converges to some real number L. now the problem is converted to this: can a sequence converge if for every m>n, [tex]|x_n-x_m|>\frac{1}{n}[/tex]? i guess such sequence doesn't converge, yet i don't know how to put it in solid proof.
can anyone help?
here is what i thought:
let's suppose the sequence is bounded, then there is its sub-sequence who converges to some real number L. now the problem is converted to this: can a sequence converge if for every m>n, [tex]|x_n-x_m|>\frac{1}{n}[/tex]? i guess such sequence doesn't converge, yet i don't know how to put it in solid proof.
can anyone help?