A question about a net in a topological space

Question

Let $X$ be any topological space and let $f:X\to \mathbb{C}$ be a bounded continuous function. Let

$\lambda = \sup \{|f(x)|:x\in X\}$.

From this, can we say that there exists a net $\{x_\alpha\}_{\alpha\in D}$ in $X$ such that

$\lim_\alpha |f(x_\alpha)|=\lambda$?

In addition, if $X$ is compact, what more we can say about it? I have begun to learning $nets$, $subnets$ in a topological space, so the question may appear trivial.

Edit The space $X$ can be assumed Hausdorff

Any help will be appreciated.

Answer 1

For each $n\in\Bbb N$ there is an $x_n\in X$ such that $\big||f(x_n)|-\lambda\big|<2^{-n}$, so clearly the sequence $\langle x_n:n\in\Bbb N\rangle$ has the property that $\lim_n|f(x_n)|=\lambda$. Thus, we can get not just a net, but a sequence.

If $X$ is compact, we can say more: in that case the sequence $\langle x_n:n\in\Bbb N\rangle$ in $X$ must have a cluster point: either it has a constant subsequence at some point $x$, or $\{x_n:n\in\Bbb N\}$ is infinite and has an accumulation point $x$. In the first case it clear that $|f(x)|=\lambda$, and in the second case continuity of $f$ ensures the same result.