I am trying to prove a step from a larger problem, and I think that it's correct because the geometry intuition that the function is decreasing on the direction that is opposite to the gradient and doesn't have minimum point. But I can't figure out how to show it, I have tried to use a methos that is similar to the gradient method combining the mean value theorem but it didn't work out.
the question is: $for f : R^n \to R, f \in C^1(R^n)$
fix $\epsilon > 0:$
$if\ ||\nabla f(x)|| > \epsilon\ \forall\ x \in R^n $
does it imply that $f$ is unbounded from below? i.e for every $M < 0 $ there exists $x_m \in R^n $ s.t $f(x_m) < M$?
thanks.