$\|f\|_p\to \|f\|_\infty$ under general assumptions I am trying to show that for nonnegative $f$ on $\mathbb{R}$ if $||f||_1<\infty$, we have
$$
\lim_{p\uparrow \infty}\|f\|_p=\|f\|_\infty
$$
I have tried to fiddle around with
$$
\left(\int_\mathbb{R}|f|^p|f|^{1-p} \right)^{1/p}<\infty
$$
but haven't found a nice way to get an estimate on $\int_E|f|^p$ from this. Honestly I am pretty uncomfortable with non finite measure spaces.
Any hints would be awesome! Any references for problems similar to this would also be awesome.
edit: this is not a duplicate of the linked question as 1) my measure space is not finite, 2) my function is in $L^1$ and not necessarily in $L^\infty$

Assume $\|f\|_\infty\lt\infty$.
For $\lambda\gt\|f\|_\infty$, $|\{x\in\Omega:|f(x)|\gt\lambda\}|=0$. Thus,
$$
\begin{align}
\|f\|_p^p
&=\int_\Omega|f(x)|^p\,\mathrm{d}x\\
&=p\int_0^\infty|\{x\in\Omega:|f(x)|\gt\lambda\}|\,\lambda^{p-1}\,\mathrm{d}\lambda\\
&\le p\|f\|_\infty^{p-1}\int_0^\infty|\{x\in\Omega:|f(x)|\gt\lambda\}|\,\mathrm{d}\lambda\\
&=p\|f\|_\infty^{p-1}\|f\|_1\tag{1}
\end{align}
$$
Therefore,
$$
\begin{align}
\limsup_{p\to\infty}\|f\|_p
&\le\lim_{p\to\infty}\left(p\|f\|_1\right)^{1/p}\|f\|_\infty^{(p-1)/p}\\
&=\|f\|_\infty\tag{2}
\end{align}
$$
If $\|f\|_\infty=\infty$, then $(2)$ is trivially true. Thus, $(2)$ is true whether $\|f\|_\infty\lt\infty$ or not. Therefore, we can remove the assumption that $\|f\|_\infty\lt\infty$.
For $\lambda\lt\|f\|_\infty$, $|\{x\in\Omega:|f(x)|\gt\lambda\}|\gt0$. Furthermore, for $\lambda\gt0$,
$$
\|f\|_p^p\ge|\{x\in\Omega:|f(x)|\gt\lambda\}|\,\lambda^p\tag{3}
$$
Therefore, for all $\lambda\lt\|f\|_\infty$,
$$
\begin{align}
\liminf_{p\to\infty}\|f\|_p
&\ge\lim_{p\to\infty}|\{x\in\Omega:|f(x)|\gt\lambda\}|^{1/p}\lambda\\
&=\lambda\tag{4}
\end{align}
$$
Since $(4)$ is true for all $\lambda\lt\|f\|_\infty$, we have
$$
\liminf_{p\to\infty}\|f\|_p\ge\|f\|_\infty\tag{5}
$$
Finally, $(2)$ and $(5)$ imply
$$
\lim_{p\to\infty}\|f\|_p=\|f\|_\infty\tag{6}
$$
