If B is a subalgebra of A, conclude that $\bar{B}$ is a subalgebra of A.
This is from Real Analysis by N. L Carothers chapter 12 exercise 3. The purpose of this is to lead up to the Stone Weierstrass theorem. I found this question but I cannot understand the answer.
The first part of this question requires us to show for f, g, h, k $\in$ A $\lvert\lvert{fg - hk}\lvert\vert$ $\leq$ $\lvert\lvert{f}\lvert\vert$$\lvert\lvert{g - k}\lvert\vert$ + $\lvert\lvert{k}\lvert\vert$$\lvert\lvert{f - h}\lvert\vert$. The second part required us to show that the multiplication operator is continuous.
Thanks again.
$fg-hk=f(g-k)+(f-h)k$. Using triangle inequality and the fact that $\|xy\|\leq \|x\|y\|$ we get $\|fg-hk\| \leq \|f(g-k)\|+\|(f-h)k\| \leq \|f\| \|g-k\|+\|f-h\| \|k\|$. To prove that the multiplication map $(f,g) \to fg$ is continuous w ehave to show that $f_n \to f$ adn $g_n \to g$ together imply $f_ng_n \to fg$. For this note that $ \|f_ng_n-fg\| \leq \|f_n\| \|g_n-g\|+\|f_n-f\| \|g\|$. Also $\|f_n\| \leq \|f_n-f\| +\|f\|<1+\|f\|$ for $n$ sufficiently large. Now can you complete the argument?