$f_k\rightarrow g$ in $L^p(\mathbb{R}^n)$ and $f_k\rightarrow h$ in $L^r(\mathbb{R}^n)$, then $g=h$ $a.e.$ in $\mathbb{R}^n$

Question

If $f_k\in L^p(\mathbb{R}^n)\cap L^r(\mathbb{R}^n) $ for some $p,r\in [1,\infty), f_k\rightarrow g$ in $L^p(\mathbb{R}^n)$ and $f_k\rightarrow h$ in $L^r(\mathbb{R}^n)$, then $g=h$ $a.e.$ in $\mathbb{R}^n$

I took $1\leq p<r$ WLOG and put $p= (1-t)1+tr$ where $t\in [0,1)$ then I applied convexity property of function $|f_k-g|^p$ but its not going too far.