Do those sequences converge?

Question

I have the following sequences and think the first 2 converge and the 3rd doesn't. Am I correct?

1. $(a_{n})=2^{-n}$ with respect to the euclidean metric.
2. in $C[0,1]$ the sequence $f_{n}(x)=\frac {x}{2^{n}}$ with metric $|f-g|=\max\{|f(x)-g(x)|:x \in [0,1]\}$.
3. $(a_{n})=2^{-n}$ with respect to the discrete metric.

Answer 1

Yes, you're right:

1) obviously converges to $0$

2)$\max\{|f_n(x)|: x\in[0,1]\}=\frac{1}{2^n}\to0$, so $f_n\to 0$ in $C[0,1]$ with the max metric

3)$d_0(2^{-n},2^{-m})=1$ for all $m,n\in\mathbb{N}$. Your sequence is not a Cauchy sequence, hence it's not convergent.