Every closed and bounded subset of $(R^n,||\cdot||_p)$ is compact $1\leq p \leq \infty$
I will show that every closed and bounded subset of the above metric space is sequentially compact, and hence compact.
Let $(x_k)_k$ be sequence in $F\subseteq (R^n,|| \cdot||_p), \; 1\leq p \leq \infty$. $F$ is closed and bounded subset of $R^n$
then, $(x_k^i)_k \; 1\leq i \leq n$ is a bounded sequence of $R$. So by Bolzano Weierstrass Theorem, there exists subsequences such that $$(x_{k_t}^i)_t \to y_i\,; \; 1\leq i \leq n$$
then, $$(x_{k_t})\to x_0$$
where $x_0 = (y_1,y_2\ldots,y_n)\in R^n$ because convergence in $(R^n,||\cdot||)$ happens only if coordinate wise convergence happens.
But since $F$ is closed, and $(x_{k_t})$ lies in $F$, so $x_0\in F$
hence, we get a subsequence of $(x_k)_k$ converging to a point in $F$
So $F$ is sequentially compact.
Is this correct?
In your argument you are picking some subsequence for each coordinate. This need not give you a subsequence of the sequence of vectors $(x_k)$. Instead, you can argue as follows. Pick a subsequence along which the first coordinates converge. The look at the second coordinates along this subsequence and extract a a further sub sequences along with the second coordinated converge. And so on.