I have the following proof in my notes:

$(C[0,1],\Vert \cdot {\Vert}_{1})$ is not complete

Consider the sequence of functions

${f}_{n}(t)=\{\begin{array}{lll}\frac{1}{\sqrt{t}}& ,& 1/n\le t\le 1\\ \sqrt{n}& ,& 0\le t\le \frac{1}{n}\end{array}$

It can easily be shown that it is Cauchy in $\Vert \cdot {\Vert}_{1}$ So we have that $\mathrm{\forall}\epsilon \mathrm{\exists}{N}_{\epsilon}$ such that $\Vert {f}_{n}-f{\Vert}_{\mathrm{\infty}}\le \epsilon \mathrm{\forall}m,n>{N}_{\epsilon}$

Suppose ${f}_{n}$ converges, Then $\mathrm{\exists}f\in C[0,1]$ such that $\Vert {f}_{n}-f{\Vert}_{1}\to 0.$

Then ${f}_{{n}_{k}}\to f\text{a.e in}[0,1]$

$f(t)=\underset{k\to \mathrm{\infty}}{lim}{f}_{{n}_{k}}(t)=\frac{1}{\sqrt{t}}\text{a.e in}[0,1]$. Absurd.

I'm just failing to understand the last line. Can someone please explain

1. what is the absurd?

2. And how does pointwise convergence follows from a.e convergence (last line). Am I supposed to understand this ${f}_{{n}_{k}}\to f\text{a.e in}[0,1]$ as convergence in the real numbers, with the absolute value for every fixed t ($|{f}_{{n}_{k}}(t)-f(t)|\text{a.e in}[0,1]$) ?