(a)
$a\ge 0,b>0.$ By definition, $a\leq f(x)\leq b\implies a^m\leq f(x)^m \leq b^m\implies \int_{0}^{1}a^mdx\leq \int_{0}^{1}f(x)^mdx \leq \int_{0}^{1}b^mdx\implies (\int_{0}^{1}a^mdx)^{\frac{1}{m}}\leq (\int_{0}^{1}f(x)^mdx)^{\frac{1}{m}} \leq (\int_{0}^{1}b^mdx)^{\frac{1}{m}}\implies a\leq c_m \leq b. $
(b)$\int_0^1 f(x)^pdx\le(\int_0^1 f(x)^\frac{pq}{p}dx)^{\frac{p}{q}}(\int_0^1 1^\frac{1}{1-p/q}dx)^{1-\frac{p}{q}}$ $=(\int_0^1 f(x)^\frac{pq}{p}dx)^{\frac{p}{q}}$ $=(\int_0^1 f(x)^{q}dx)^{\frac{p}{q}}$ We know, $\frac{p}{q}<1$ so $c_n$ is increasing. We know that $\lim_{p\to\infty}||f||_{p}=||f||_{\infty}$ so $c_n$ converges to $b$. Am I right?