Which of the following are true for the sequence $f_n(x)=(-x)^n$ for $x\in[0,1]$

Question

Which of the following are true for the sequence $f_n(x)=(-x)^n$ for $x\in[0,1]$

1. there exists a pointwise convergent subsequence of $f_n$.
2. $f_n$ has no pointwise convergent subsequence.
3. $f_n$ converges pointwise everywhere.
4. $f_n$ has exactly one pointwise convergent subsequence.

By Bolzano–Weierstrass theorem, option 1 is true. Also for $x=1$, $f_n(x)=(-1)^n$ which is not convergent. so option 3 is not true. Also option 2 is false. But I am not sure about the option 4. How can I check this option? Any hint or help would be great. Thanks.

Answer 1

Hint: Consider $f_{2n}$ and $f_{2n+1}$.