Borel-Cantelli lemma (a fixed string of zeros and ones of length $k$)

Question

I want to show the probability that a fixed string of head and toss of length $k$ appears when I flip a coin, is one.

I think to use a binary expansion of $\omega$ for $\omega \in [0,1]$;

I define $d_n(\omega)$ the nth binary digit and $a_i$ the digit of the string

So I have the event $$A_n=\{d_n(\omega)=a_i,......,d_{n+k-1}(\omega)=a_{i+k-1}\}$$

The teacher says the $A_n$ aren't independent so I can't use Borel-Cantelli Lemma 2.

But why aren't they independent?