For $1<p<q<\infty$ show $L^q[0, 1] \subset L^p[0, 1]$

Question

For $1<p<q<\infty$ show $L^q[0, 1] \subset L^p[0, 1]$

I know I'm supposed to use holders inequality to solve this, but can't work it out. We haven't learnt anything about measure theory, so all the answers I've come across previously using measures haven't made sense.

Answer 1

There's a very nice proof without Holder. Take $f \in L^q([0,1])$ and write

$\int_0^1 |f(x)|^pdx = \int_{|f|<1,x\in[0,1]} |f(x)|^pdx+\int_{|f|\geq1,x\in[0,1]} |f(x)|^pdx.$

Then use the hypotheses to bound the RHS.