How a linear function is bounded below on $R^n$ only when it is zero and not -$\infty$?

Question

I’ve gone through these two links

Why is a linear function bounded below on $R^n$ only when it is zero?

Bounded linear function implication

But I’m not able to understand why there is a bound. Can’t a function be unbounded from below? According to 2nd link the author has considered a ray. What if it’s a line with range $R$ , as function is defined as $f : \mathbb{R}^m \to \mathbb{R}$ , hence large -ve input value means large -ve Output, how it’s bounded by zero?

Extremely sorry it’s difficult to digest.

Answer 1

I think you have misunderstood the question: the point is that the zero function is the only linear function on $\mathbb{R}^n$ which is bounded below: that is, any non-zero linear function is not bounded below, as you say.

Answer 2

Suppose $f(x)= y \neq 0$ for some $x\in\mathbb{R}^n$.

Now suppose we have a bound $B$ such that we know $f(a) > B$ for all $a\in\mathbb{R}^n$. But by linearity we have $f(\frac{B}{y}x) = \frac{B}{y}f(x) = \frac{B}{y}y = B$, which is a contradiction.