Prove a inequality about integral and summation

Question

If $f(x)$ is monotonic increasing on the interval $a\leq x < \infty$, could we prove following inequality formally?

\begin{equation} f(a+k) \leq \int_{a+k}^{a+k+1} f(t) dt \leq f(a+k+1) \end{equation}

Answer 1

Hint: $f(a+k) \leq f(t) \leq f(a+k+1)$ for $t \in [a+k, a+k+1]$

Answer 2

Write \begin{align} f(a+k) &= \inf\{f(u) \mid u \in [a+k,a+k+1) \} \\ &= \int_{a+k}^{a+k+1} \inf\{f(u) \mid u \in [a+k,a+k+1) \} dx \\ &\leq \int_{a+k}^{a+k+1} f(x) dx \\ &\leq \int_{a+k}^{a+k+1} \sup\{f(u) \mid u \in [a+k,a+k+1) \} dx \\ &= f(a+k+1). \end{align}