# Thread: Limit points of subsequences

1. ## Limit points of subsequences

I have to give examples of four kinds of subsequences if they exist. S∈ℝ is the limit point of a subsequence of the sequence (Xn) (n=1 -> ∞) if ∃ subsequence (Xnk) (k=1 -> ∞) : Xnk -> S.

What could the sequences be or do they exist if

a) limit point of subsequence = {0}
b) limit point of subsequence = {0,1}
c) limit point of subsequence = {infinitely many points}
d) limit point of subsequence = Q ∩ [0,1]

2. Originally Posted by antero
I have to give examples of four kinds of subsequences if they exist. S∈ℝ is the limit point of a subsequence of the sequence (Xn) (n=1 -> ∞) if ∃ subsequence (Xnk) (k=1 -> ∞) : Xnk -> S.

What could the sequences be or do they exist if

a) limit point of subsequence = {0}
Do you mean by this that the only subsequential is 0? Any sequence converging to 0 will do.

b) limit point of subsequence = {0,1}
Find a sequence, $\{a_n\}$, that converges to 0, a sequence, $\{b_n\}$, that converges to 1, and "interleave" them: $a_1, b_1, a_2, b_2, ...$.

c) limit point of subsequence = {infinitely many points}
It is a property of all real numbers, that, given any $\epsilon$, there exist a rational number within distance $\epsilon$ of the real number. Since the rational numbers are countable, we can order them: $r_1, r_2, r_3,\cdot\cdot\cdot$. That is, the set of all rational numbers form a sequence having all real numbers as subsequential limits, and there are, of course, an infinite number of them.

d) limit point of subsequence = Q ∩ [0,1]
Since there are an infinite number of rational numbers in 0, 1, any example for d is also an example for c. But, unfortunately, the example I gave for c does not work since its set of sequential points is too big- it includes much more than the rational numbers. I don't think you will be able to give a specific, numerical, example. You might do something like this:
Since the set of rational numbers in [0, 1] is countable, it is possible to put them into a sequence: $r_1, r_2, r_3, \cdot\cdot\cdot$. It is a property of all real numbers, and in particular of rational numbers, that given any rational number, $r_i$, there exist a sequence of rational numbers, [tex]\{a_{ij}\}[tex] that converges to that rational $r_i$. Now "interleave" those: $\{a_{11}, a_{12}, a_{13}, \cdot\cdot\cdot\, a_{21}, a_{22}, a_{32}, \cdot\cdot\cdot, a_{31}, a_{32}, a_{33}, \cdot\cdot\cdot\ \}$.