A question about Cauchy sequence

Question

Let $\{a_n\}$ be a Cauchy sequence Then

1. Show that $\{a_n\}$ is bounded
2. Show that $\{a_n\}$ is convergent
3. Show that there is at least one subsequential limit point of $\{a_n\}$
4. Show that there is no more then one subsequential limit of $\{a_n\}$

for (1)

Let $\{a_n\}$ be Cauchy sequence

taking $\epsilon =1 $ there exists a positive integer $m$ such that

$a_n-a_m<1\;\; \forall n \ge m$

then $a_m-1<a_n<a_m+1$

$K=min\{a_1,a_2,......,a_{m-1},a_m-1\}, K=max\{a_1,a_2,......a_{m-1},a_m+1\}$

then $k \le a_n \le K$ this means $\{a_n\}$ is bounded

for (2) convergent iff cauchy

how to prove (3) and (4)

Answer 1

For (2), I would be surprised that you can use the fact that a Cauchy sequence converge (the aim is at my opinion to show that a Cauchy sequence converge).

(3) It's Bolzano-Weierstrass theorem

(4) If there is two limits point or more, you will have a contradiction with (2).