Question About the Relationship Between Pointwise Supremum and Pointwise Limit

Question

I encounter a simple question arising from basic real analysis: Consider a sequence of functions, call it $\{f_k\}$, defined as $$ {f_k}\left( x \right): = \left\{ \begin{gathered} 0,\begin{array}{*{20}{c}} {}&{ - 1 \leq x \leq 0} , \end{array} \hfill \\ kx,\begin{array}{*{20}{c}} {}&{0 < x < \frac{1}{k}} , \end{array} \hfill \\ 1,\begin{array}{*{20}{c}} {}&{x \geq \frac{1}{k}}. \end{array} \end{gathered} \right. $$ Then I know the corresponding pointwise limit, call it $f(x)$, is $$ f(x):=\lim_{k \to \infty} {f_k}\left( x \right) = \left\{ \begin{gathered} 0,\begin{array}{*{20}{c}} {}&{ - 1 \leq x \leq 0}, \end{array} \hfill \\ 1,\begin{array}{*{20}{c}} {}&{x > 0} . \end{array} \hfill \\ \end{gathered} \right. $$ However, suppose I change my mind and say I want to ask what is the pointwise supremum over the sequence of functions $\{f_k\}$, i.e., if I compute $\sup_k f_k(x)$, then I think this pointwise supremum is identical to the pointwise limit. In short, I think I could obtain $$ \sup_k f_k(x) = f(x) $$ Am I saying all this correct? If so, is this property (the pointwise supremum is identical to the pointwise limit) just a coincidence for this specific example, or it has a more general setting which I could explore. Any comments are appreciated.

Answer 1

You are asking a basic questions about sup and inf of sets of real numbers. For each $x$ you have the set $\{f_k(x):k\in\mathbb{N}\}$. The pointwise limit is just $\lim_{k\rightarrow\infty} f_k(x)$. The pointwise sup is just the sup of each of these sets. Since the example you gave has the sets increasing and bounded for each $x$ then the sup will be the same as the limit. You could also consider the pointwise inf or limsup or liminf.

In general, these will not be the same thing. (eg. consider $-f(x)$ for your above example. The pointwise inf will then be the same as the pointwise limit but the pointwise sup will be different).