Apologies if this is to obvious of a question; for some reason I'm having a tough time thinking about this.
Suppose $f$ and $g$ are continuous, monotonically decreasing functions from $\mathbb{R}$ to $\mathbb{R}$. Further suppose that $f$ intersects $g$ from above; denote the $x$ at which they intersect by $x^*$ (assume for simplicity that it is unique). Now, for some $\omega>0$, define $\tilde{f}(x):=f(x-\omega)$ (i.e. rightward shift by amount $\omega$).
I am trying to prove that $\tilde{f}$ and $g$ intersect at $\tilde{x}^*>x^*$, but for some reason I am finding this difficult. I would appreciate any guidance you can provide.