In a course of writing a paper I realised that I need a lemma below, and found a half page proof. I have not seen this lemma before and wonder if maybe someone here knows this lemma or can prove it in five lines. Please let me know if this is the case.
Lemma. Let $C$ be an annulus in $\mathbb R^2$ given by inequalities $1\le x^2+y^2\le 2$. Let $f$ be a smooth function on $C$, whose gradient vanishes only in the interior of $C$ and whose only critical points are isolated minima. Then there is a gradient trajectory of $f$ that joins two boundaries of the annulus.
