To show a function is in $L^r(\mu)$

Let $(X,\mu)$ be a finite measure space and $f:X\to [-\infty,\infty]$ be such that $$\mu\{|f|\geq \lambda\}\leq C \frac{1}{\lambda^p}, \text{for some } 1<p<\infty.$$ for some positive constant $C$. How to show that $f\in L^r(\mu), 1\leq r<p?$

I tried to split the integration for $|f|<\lambda$ and $|f|\geq \lambda$ but I could not manage to show finite the second part of integration.

Let $1\le r<p$. I let you check that \begin{align*} \int|f|^r\,\mathrm d\mu&=\int_0^\infty\mu(|f|^r\ge\lambda)\,\mathrm d\lambda\\[.4em] &\le\mu(\Omega)+\int_1^\infty\mu(|f|\ge\lambda^{\frac1r})\,\mathrm d\lambda\\[.4em] &\le\mu(\Omega)+\int_1^\infty\frac C{\lambda^{\frac pr}}\,\mathrm d\lambda\\ &<\infty. \end{align*}