While studying functional analysis, more specifically that the subspace $Y=\{x \in \mathcal{C}[a, b] \mid x(a)=x(b)\} \subset \mathcal{C}[a, b]$ is complete, I came across a very simple question I cannot seem to answer, and I am afraid I am missing something simple here.
To show that the limit of the sequence, which is described in the related link, belongs to $Y$, one should show that $x(a)=x(b)$. But why is that enough? Shouldn't one also show the limit of the sequence belongs to $\mathcal{C}[a, b]$? In my opinion this is missing, since it is also a requirement for $x$ to belong to $Y$.
Thanks in advance, Lucas
Asserting that $\lim_{n\to\infty}x_n=x$ with respect to the $\sup$ norm is the same thing as asserting that $(x_n)_{n\in\Bbb N}$ converges uniformly to $x$. And uniform convergence preserves continuity.