I started a course on Analysis last month, and we have a question "How many accumulation points could Cauchy sequence possibly have?". I will take this example:
$$a_n= ({1 + {1\over n}})^n$$
And examine it in it metric space $\mathbb Q$.
From the textbook I see that every Cauchy sequence in metric space X is bounded, therefore by BW theorem it must have at least one accumulation point(in this case it's $e$ ). But when I go back in my textbook I see that accumulation point has to be the element of given metric space X (in this case it's $\mathbb Q$). It's obvious that this sequence is Cauchy, but I can't see that it has accumulation point in $\mathbb Q$.
So can someone explain to me what am I doing wrong?
Yes, and your example is a good exemple. Your sequence is strictly increasing and do not converge, therefore it can't have accumulation point. But if the space is complete (for example $\mathbb R$) it's impossible since in a complete space every Cauchy sequence converge (and thus has a unique accumulation point).
Notice that $$\text{BW true } \iff\text{ the space is complete},$$ since $\mathbb Q$ is not complete, BW is not useable.