Uniform convergence of $f_n(z)=\sum_{j=o}^n z^j$ on the open unit complex disk.

Question

I have the sequence $f_n(z)=\sum_{j=o}^n z^j$ on the open unit complex disk ($\Delta$).

My question is whether or not my approach is correct to the following problems:

1. is the sequence normal?
2. Does the sequence converge pointwise on $\Delta$ to a limit $f$?
3. Is the convergence $f_n\to f$ locally uniformly in $\Delta$?
4. Is the convergence $f_n\to f$ uniform on $\Delta$?

My attempt:

1. Yes the sequence is normal as on any compact subset of $\Delta$, $|z|\le K\lt 1$, so the sequence (which converges to the geometric series) converges uniformly (and thus there is a subsequence converging uniformly on compat subsets of $\Delta$).

2. Yes, the pointwise limit is $\frac{1}{1-z}$ (again since the sequence converges to the geometric series for $|z|\lt 1$)

3. Yes, for any $x\in\Delta\exists r\gt 0$ such that $z\in B(x,r)\Rightarrow |z|\le K\lt 1$ thus in this ball $f_n\to f$ uniformly.

4. No. Since if so, the sequence would be uniformly cauchy convergent as well. But uniform cauchy convergent gives us:

$\forall\epsilon\gt 0,\exists N\in\Bbb N:\forall m,n\ge N :|f_n(z)-f_m(z)|\lt\epsilon,\forall z\in\Delta$.

If this were to hold, then it would hold for $n=m+1,\forall n,m\ge N$ and we have:

$|f_n(z)-f_m(z)|=|\sum_{j=0}^nz^j-\sum_{j=0}^mz^j|=|\sum_{j=0}^{m+1}z^j-\sum_{j=0}^mz^j|$

$=|z^{m+1}|=|z|^{m+1}\le |z|^{N+1}\le|z|^N\lt\epsilon$

But such $N$ would depend on $\epsilon$ and $z$, so the condition is not uniform.

I would really like clarification on whether what I have done is correct, and if so I would also appreciate advice on how to write the final statement "But such $N$ would depend on $\epsilon$ and $z$, so the condition is not uniform." more rigorously if possible.

Thanks very much!