relationship between $f_n$ and $f$

Question

So when looking at functional sequences in terms of uniformally convergent, I am struggling to translate the definitions into examples.

I understand that $f_n=\frac{x^2}{n}$ is the sequence $f_n$={$x^2$, $\frac{x^2}{2},$ $\frac{x^2}{3}...$}

However how does $f$ relate to $f_n$. If i have a definition that says: |$f_n(x)-f(x)$| and $f_n$ is the above sequence then what is $f(x)$

Thanks

Answer 1

In fact, the sequence $(f_{n})$ should be written $$ (x \mapsto x^{2}/1, x \mapsto x^{2}/2, x \mapsto x^{2}/3, \dots ) $$ instead; note the difference between a function and the value of a function at a point, which matters here.

The limit function (if exists) of a sequence of functions is also a function; for example, if $f_{n}: x \mapsto x^{2}/n$ on $[0,1]$ for each $n \geq 1$, then $|f_{n}(x)-0| = x^{2}/n \leq 1/n \to 0$ as $n$ grows indefinitely for all $x \in [0,1]$, implying $(f_{n})$ converges uniformly to the function $x \mapsto 0: [0,1] \to \Bbb{R}$ (usually abbreviated as "$(f_{n})$ converges uniformly to $0$").

Answer 2

For every $x$ separately it is clear that $f_n(x)$ tends to 0 so presumably in this example $f$ would be the constant 0 function. The sequence may or may not converge uniformly depending on the domain chosen (bounded or unbounded)