Prove that if $f$ is of bounded variation on $[a, b]$, then $f$ is bounded on $[a, b]$.

Question

Prove that if $f$ is of bounded variation on $[a, b]$, then $f$ is bounded on $[a, b]$.

---

My professor proved the proposition like the following processes:

1. Choose $x$ between $a$ and $b$, that is, $x\in(a, b)$.

2. let $\Gamma$ be a partition having elements $a$, $x$ and $b$, that is, $\Gamma = \{a, x, b\}$.

3. For a constant M, $$S_\Gamma[f; a, b] \le V[f; a, b] \le M \lt +\infty$$

4. By the definition of $S_\Gamma$ \begin{align} S_\Gamma[f; a, b] &= |f(x)-f(a)| + |f(b)-f(x)| \\ &\ge |f(x)| - |f(a)| + |f(x)| - |f(b)| \\ &= 2|f(x)| - |f(a)| - |f(b)| \end{align}

5. Therefore, \begin{align} |f(x)| &\le \frac{S_\Gamma[f; a, b]+|f(a)|+|f(b)|}{2} \\ &\le \frac{V[f; a, b]+|f(a)|+|f(b)|}{2} \\ &\le \frac{M+|f(a)|+|f(b)|}{2} \\ &\lt +\infty \end{align}

6. Therefore, $f$ is bounded on $[a, b]$ since $|f(x)|\lt +\infty$

---

I think that this proof pre-assumed that $|f(a)|$ and $|f(b)|$ are bounded. However, I cannot help casting doubt on this pre-assumption. Shouldn't I deal with the case that either $f(a)$ or $f(b)$ is either $+\infty$ or $-\infty$.