What theorem said that $L^2[a,b]$ is dense in $L^1[a,b]$ in Royden "Real analysis " 4th edition.

Question

I was reading the solution of this question:

Show that $L^2[a,b]$ is of the first category

But I do not know: What theorem said that $L^2[a,b]$ is dense in $L^1[a,b]$ in Royden "Real analysis " 4th edition. Could anyone help me in finding this theorem ?

Also, could anyone tell me the general idea of the solution given in this link please?

Answer 1

If $f \in L^{1}[a,b]$ then $f_n =fI_{\{|f| \leq n}\}$ is sequence in $L^{2}[a,b]$ converging in $L^{1}[a,b]$ to $f$.