Sequence of integrable functions on [a,b] converges pointwise but not uniformly?

Question

I'm currently taking Real Analysis and there was an example in the textbook showing that the sequence of integrable functions fn(x) converges pointwise. I'm a bit confused as to why that is and also why does it not converge uniformly? Here is the example in textbook!

Thank you!

Answer 1

A sequence $f_n$ converges uniformly to $f$ if for all $\epsilon > 0$, there exists $N$ such that if $n \geq N$, $\vert f_n(x) - f(x)\vert \leq \epsilon$ for all $x \in [a,b]$.

In the example, the $f_n$ converge pointwise to zero because for every $x$ there exists $N$ such that $x > 1/N$. If a uniform limit exists is certainly must agree with the pointwise limit (uniform convergence is a type of pointwise convergence). However, $f_n$ has a point with value $n$ for all $n$.

Answer 2

Because it converges pointwise to the null function. If the convergence was uniform, then, by the definition of uniform convergence, there would be a natural $N$ such that$$(\forall n\in\mathbb N)(\forall x\in[0,1]):n\geqslant N\implies\bigl\lvert f_n(x)\bigr\rvert<1,$$and this cleary does not occur.