# How to prove that $L^q$ space is a subset of $L^p$ space if $1 \leq p < r < q < \infty$? [duplicate]

How to prove that $L^p$ space is a subset of $L^q$ space whenever $1 \leq p \leq q < \infty$?

I tried to solve it using the $L^p$ space definition. Tell me how to proceed further?

## marked as duplicate by tomasz, user91500, user370967, Davide Giraudo, kingW3Jun 23 '17 at 11:20

This isn't true. Consider $f(x) = \frac 1{\sqrt x}$ which is in $L^1[0,1]$ but not in $L^2[0,1]$ .