I want to prove that any closed and bounded set in $\mathbb{R}^n$ is sequentially compact.
Note that I want to prove this without using the Heine-Borel theorem and the equivalence of sequential compactness and compactness. Ah and I can use that any closed and bounded set in $\mathbb{R}$ is sequentially compact.
The way I did it is as follows: Let $S$ be a closed and bounded subset of $\mathbb{R}^n$. Then $\exists a,b \in \mathbb{R}$ such that $S \subseteq [a,b]\times [a,b] \times ...[a,b]$. Now I will consider a sequence in $S$.
Let $x_p = (x_p^1, x_p^2,...,x_p^n) \forall p \in \mathbb{N}$.
Then I will construct $n$ sequences of components of $x_p$. Let $x^{(i)}=x_1^i,x_2^i...$ Then $x^{(1)}\subset [a,b]$, and $[a,b]$ is sequentially compact. so $x^{(1)}$ has a convergent subsequence. Now if we denote the indices of this sequence by $t_1,t_2...$ we know that $x^1_{t_1},x^1_{t_2}, x^1_{t_3}...$ converges to some $x$ in $[a,b]$. Now if we consider $x^2_{t_1},x^2_{t_2}, x^2_{t_3}...$ it is a sequence in $[a,b]$ a sequentially compact space. So it has a convergent subsequence. Again we take the indices of this subsequence and we consider them to build a subsequence of $x_3$ which converges.
After we have repeated this procedure $n$ times we are left with $n$ we have found a set of indices $f_1,f_2...$ such that $x^n_{f_1}, x^n_{f_2}...$ converges. But then by construction $x^p_{f_1}, x^p_{f_2}...$ converges $\forall p \in1..n$ as it is a subsequence of a convergent sequence.
Then $x_{f_1},x_{f_2} ..$ converges (as it is a sequence in $\mathbb{R}^n$ for which all individual components converge) and as it is included in a closed set, the limit of the sequence is part of the set.
Since we started with an arbitrary sequence, this proves that any closed and bounded set in $\mathbb{R}^n$ is sequentially compact.
Can you please have a look at this and tell me if something is wrong? It seems fine to me, but I really want to get the concept behind sequential compactness. I know that historically it was the first definition of compactness, so I find it important to see how knowledge evolved and how people initially proved it, without using compactness.
Also, please feel free to show me another proof, I am very interested in that as well.
Thank you!