Solving an Infinite Sequence: Convergence to \alpha

[ptolema]

Homework Statement

consider the sequence 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, ...
for which numbers $$\alpha$$ is there a subsequence converging to $$\alpha$$?

Homework Equations

none that i can think of...

The Attempt at a Solution

i think the $$\alpha$$ values are 1 and 0, but I'm not sure how i can tell if those are the only numbers that subsequences can converge to. what kind of method should i use to determine the values. to get 1 and 0, i just kind of stared at the numbers for a bit, but i'd like to refrain from doing that.

 

[Dick]

Homework Statement

consider the sequence 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, ...
for which numbers $$\alpha$$ is there a subsequence converging to $$\alpha$$?

Homework Equations

none that i can think of...

The Attempt at a Solution

i think the $$\alpha$$ values are 1 and 0, but I'm not sure how i can tell if those are the only numbers that subsequences can converge to. what kind of method should i use to determine the values. to get 1 and 0, i just kind of stared at the numbers for a bit, but i'd like to refrain from doing that.

Ok, try and retarget your staring. Try and find a number alpha in [0,1] that such that there is no subsequence converging to alpha.

 

[ptolema]

so, i can see the sequence converging to 1, 0, and rational numbers in [0,1]. can this series converge to irrational numbers, as well?

 

[Dick]

so, i can see the sequence converging to 1, 0, and rational numbers in [0,1]. can this series converge to irrational numbers, as well?

The series doesn't converge to anything. Subsequences can. Of course, they can converge to an irrational. You make sequences of rationals converging to irrationals all the time to approximate them. Have you noticed your sequence contains ALL rational numbers in (0,1)? And it, in fact, contains each one an infinite number of times?

 

[ptolema]

sorry, the subsequences is what i meant, mixed up wording. i guess when you think about it, you can make infinitely many subsequences, and they can converge to any value on [0,1]

 

[Dick]

sorry, the subsequences is what i meant, mixed up wording. i guess when you think about it, you can make infinitely many subsequences, and they can converge to any value on [0,1]

That's not really a proof or anything. But it's certainly true.

 

[ptolema]

thanks. i guess i have to puzzle out the proof now, wish me luck!

 

[Dick]

thanks. i guess i have to puzzle out the proof now, wish me luck!

Wish you luck!