Vtyjkyuk

This is my proof refer to baby rudin Thm 2.43.
Let $P$ is perfect set, then $P$ is infinite set.(Because P don't have any isolated point.)
If $P$ is countable $P$ should be represented by
$$P=x_1,x_2,cdots $$
Now labeling following way, Because P is non-empty
,$exists x_1in P$,

Let a arbitrary neighborhood $N_r_1(x_1)$.

Since every point in $P$ is limit point of $P$.
$exists x_2 in P$.

Let $r_2=frac12min(d(x_1,x_2),r_1-d(x_1,x_2))$

Then $(i)barN_r_2(x_2)subset N_r_1(x_1)$, ($barN$ imply closure of N)
,$(ii) x_1notin N_r_2(x_2)$,$(iii)N_r_2(x_2)cap Pneq emptyset$
similar way we can make sequence $N_r_n(x_n)$ satisfying

$(i)barN_r_n+1(x_n+1)subset N_r_n(x_n)$
,$(ii) x_nnotin N_r_n+1(x_n+1)$,$(iii)N_r_n+1(x_n+1)cap Pneq emptyset$

Let $K_n:=barN_r_n(x_n)cap P$, then P is closed set, and $barN_r_n(x_n)$ is closed and bounded, so $K_n$ is closed subset of compact set, therefore, $K_n$ is compact.

$forall nin N$, $K_n$ is non-empty compact set satisfying $K_n+1in K_n$,
so $bigcap_nin NK_nneq emptyset.$
Otherwise, $x_nnotin K_n+1$, so there should be exist $yneq x_n, forall nin N$, $yinbigcap_nin NK_n$. $P$ is not represented by $x_n$, so P is uncountable.

Here is question,
I don't know what is difference between above proof and following,

For $x_1in N$, every sequence,$x_n+1=x_n+2$ is not represented $N$ perfectly, so $N$ is uncountable.

what is the difference of two proofs?, or If my proof for" every perfect set is uncountable." is not exact, where is exact proof?