If $f:[a,b]\rightarrow \mathbb{R}$ is Riemann integrable does $\lim_{p\rightarrow\infty}(\int_a^b|f(x)|^p)^{\frac{1}{p}}=\sup_{x\in[a,b]}|f(x)|$?

Question

Several posts on this site ask for a proof of the statement

If $f:[a,b]\rightarrow\mathbb{R}$ is continuous, $\lim_{p\rightarrow\infty}(\int_a^b|f(x)|^p)^{\frac{1}{p}}=\sup_{x\in[a,b]}|f(x)|$.

Need $f$ only be Riemann integrable for $\lim_{p\rightarrow\infty}(\int_a^b|f(x)|^p)^{\frac{1}{p}}=\sup_{x\in[a,b]}|f(x)|$ to be true?

If so, where can I find a proof of this that does not involve measure theory?