In a few hours I will have a quiz and while studying I had some questions. Could you please help me?
Thanks in advance
Question 1: Let $(X, d)$ be a metric space and $(x_n)$ be a sequence in $X$. Let $(x_n)$ be a Cauchy sequence, prove that if $(x_n)$ has a cluster point $z$, then $(x_n) \to z$.
My attempt: Suppose that $z$ is a cluster point of $(x_n)$, and construct a subsequence $x_{n_k}$ with limit $z$.
Choose $x_{n_1} \in B(z,1)$, $B(z,1/2)$ contains infinitely many elements of $(x_n)$
Choose $x_{n_2} \in B(z,1/2)$ such that $n_2 > n_1$
..... ..... .....
Choose $x_{n_k} \in B(z,1/k)$ so that $n_k > n_{k-1} > \cdots$
By the choice we did, $d(x_{n_k},z) < 1/k$ and we know that $1/k \to 0$ then, $d(x_{n_k},z) \to 0$. Hence, $x_{n_k} \to z$.
Now how can I show that $x_n$ also converges to $z$?
Question 2: Let $a_n > 0$, $b_n > 0$, $n=1,2,\dots$. Prove that if $\frac{a_n}{b_n} \to l \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$ and $\sum _{n=1}^{\infty} b_n$ converge or diverge simultaneously.
My attempt: If $\frac{a_n}{b_n} \to l \neq 0$ then we can also say that if $\frac{a_n}{b_n}$ is closed by $l$.
If $l>0$, then there are positive numbers $c$ and $d$ with $0
Now if $\sum_{n=1}^{\infty} b_n$ converges, then so does $\sum_{n=1}^{\infty} db_n$. From the inequality $\frac{a_n}{b_n} \leq d$, $\sum_{n=1}^{\infty} a_n$ also converges. However if $\sum_{n=1}^{\infty} b_n$ diverges, then so does $\sum_{n=1}^{\infty } cb_n$.
From $a_n \leq db_n$ we see that if $\sum_{n=1}^{\infty} a_n$ diverges then $\sum_{n=1}^{\infty} \frac{a_n}{d}$ also diverges and in this case $\sum_{n=1}^{\infty} b_n$ diverges too. With similar argument we can see that if $\sum_{n=1}^{\infty} a_n$ converges then $\sum_{n=1}^{\infty} \frac{a_n }{c}$ also converges and from the left inequality ($b_n < \frac{a_n}{c}$) $\sum_{n=1}^{\infty} b_n$ converges too.
Is this proof right and enough?