Suppose that $f(x)$ is a real valued continuous function such that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$. Further, let
\[a_{n}=\sum_{j=1}^{n} 1 / j\]
and
\[b_{n}=\sum_{j=1}^{n} 1 / j^{2} .\]
Which of the following statements is true? ($\text{Hint:}$ Consider the convergence properties of the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$.)
- There exists a number $M$ and a positive integer $n_{0}$ so that $f\left(a_{n}\right) \leq M$ and $f\left(b_{n}\right) \leq M$ for all $n \geq n_{0}$
- There exists a number $M$ and a positive integer $n_{0}$ so that $f\left(a_{n}\right) \geq M$ and $f\left(b_{n}\right) \leq M$ for all $n \geq n_{0}$
- There exists a number $M$ and a positive integer $n_{0}$ so that $f\left(a_{n}\right)-f\left(b_{n}\right) \leq M$ for all $n \geq n_{0}$
- There does not exist any number $M$ so that $f\left(b_{n}\right)$ and $f\left(a_{n}\right)$ are greater than $M$ for all $n$
- None of the above