Show that there is an uncountable number of Cauchy sequences of rational numbers equivalent to any given Cauchy sequence of rational number

Question

Please check my proof :)]

We must show if given $\epsilon >0$ we can find n>N such that $|S_{n}-L_{n}|<\epsilon $

We set up $|S_n-L_n|$ for any uncountable cauch sequence as

$$|S_{1}-L_{1}|<\frac{\epsilon }{2}$$

and for any cauchy sequence of rational number as

$$|S_{2}-L_{2}|< \frac{\epsilon }{2}$$

then

$$|S_{n}-L_{n}| = |S_{1}-L_{1}+S_{2}-L_{2}|\leq |S_{1}-L_{1}|+|S_{2}-L_{2}|\leq \frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon $$

Answer 1

Hint: Let's say that you have convergent sequence $(a_n)$. Could you find uncountably many sequences $(b_n)$ such that $\lim_na_n=\lim_nb_n$? To further hint toward solution, what do you know about subsequences of convergent sequence and how many of them are there?

Cauchy sequences of rational numbers converge to real numbers, so if you can answer the above question, you are done.