Questions about set cardinalities (infinite/finite).

I have a couple questions that I undertstood (maybe) intuitively and tried to prove them somehow, I would like if someone can give feedback and tell me if I’m doing it the right way.
True or false:
1)$ |{ frac {1} {2^n} mid nin mathbb{N} }| = |mathbb{N}|$
.
Intuitively this seems true, and I thought of proving it by introducing the function $f: mathbb{N} longrightarrow { frac {1} {2^n} mid nin mathbb{N} }$, $f(n)=frac {1} {2^n}$.

2) $K= {Ain P(mathbb{N}) mid A space is space finite } $, Assumption: $K$ is finite.

Intuitively I can see it’s infinite, since the power set of natural numbers have infinity elements, but I really don’t know how to prove it.

3) $L= {Ain P(mathbb{N}) mid A^c space is space finite }$,
Assumption $L$ is finite.
I think this is true, since for all $Ain P(mathbb{N})$, $A^c$ is infinite, so theres no elements in $L$.

I would really appreciate it if someone can approve my work, and tell me how to formalize the proofs, and point out my mistakes. Thanks in advance