Can a Sequence in R Lack Convergent Subsequences or Target Specific Limits?

[yifli]

1. a sequence in R having no convergence subsequence

2. nonconvergent sequence in R such that the set of limit points of convergent subsequence consists exactly of the number 1

3. a sequence x_n in [0, 1] such that for any y in [0,1] there is a subsequence $$x_{n_m}$$ converging to y

 

[HallsofIvy]

(1) is pretty close to trivial. Give it another try.

(2) Can you find a sequence that converges to 1? Combine that and your answer to (1) by letting all even terms be from one sequence and all odd terms be from the other.

Now (3) looks hard!

 

[kru_]

For 3, this is a classic example and you can probably come up with a good example just by rewording the question.

If you want the hint, think about this, what does the sequence $$\frac{1}{1},\frac{1}{2},\frac{2}{2},\frac{1}{3},\frac{2}{3},\frac{3}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{4}{4},\frac{1}{5},\frac{2}{5}\ldots$$ converge to?

 

[JG89]

For 3) you can also use the fact that the countable union of countable sets is itself countable and that for any x in [0,1] there is a sequence of rationals converging to x.