"if K is compact in R^p, then K x {a} is also compact in R^(p+1)" withough using Heine-Borel Theorem

Question

Should we prove the fact "if K is compact in R^p, then K x {a} is also compact in R^(p+1)" ???

If so, Can We prove "if K is compact in R^p, then K x {a} is also compact in R^(p+1)" by using only definition of compactness ( in other words, by only using the fact that K is compact if every covering A of K can be replaced by a finite covering of K, using only sets in A )...

(I am studying compactness in analysis(book: Elements of Analysis - Robert G bartle), not topology... So It will be helpful not to use any topological knowledge if you can prove...)

Answer 1

Have you been taught that in $R^p$,compactness is the same as being closed and bounded? p.s. I didn't notice that you couldn't use Heine-Borel. In that case take any open cover $U_\alpha$ of $K\times(a)$. $U_\alpha\cap R^p$ will be an open cover of K.Choose a finite cover,and then at the most add one more open set to cover $K \times (a)$.

Answer 2

If you take a sequence of the form $x_n\times a, x_n\in K$, you can find a converging subsequence $x_{n_k}$ of $x_n$ since $K$ is compact, $x_{n_k}\times a$ is a converging subsequence of $x_n\times a$.