I'm reading about Abel's limit theorem, and I don't get why the result isn't practically trivial. Below, I will state the theorem and my very, very simple argument. I would appreciate if someone could let me know where I'm going wrong.
$\mathbf{Theorem}$: If $\sum_{n=1}^{\infty} a_{n}$ exists, then $\sum_{n=1}^{\infty} a_{n}x^{n}$ converges uniformly for $x \in [0,1]$.
"$\textit{Proof}$" (incorrect): Obviously, the key thing to prove above is that we can include the point $x = 1$ in the interval on which the series converges uniformly. For $x < 1$, uniform convergence follows from the theorem on power series. Now, my bad argument goes as follows:
We know that there is $N_{1} \in \mathbb{N}$ s.t. $\sum_{n=N_{1}}^{\infty} a_{n}x^{n} < \varepsilon$ for $x <1$.
Additionally, we have $N_{2} \in \mathbb{N}$ s.t. $\sum_{n=N_{2}}^{\infty} a_{n} < \varepsilon$ (i.e. the above power series with $x=1$).
From this, why is it incorrect to conclude that we have uniform convergence on $[0,1]$ by picking $N \geq $max$(N_{1}, N_{2})$, i.e. why do we need Abel's theorem?