Proving properties of a spring-mass system A spring-mass system is governed by $$u'' + u = \cos(\omega t), \hspace{10mm} u(0)=u'(0)=0,$$ Where $\omega \geq 2$ is a given constant. a.) Find $u(t)$ b.) Show that $|u(t)| \leq 1$ for all $t$ c.) Can you find a constant $A < 1$ such that $|u(t)| \leq A$ for all $t$? How small an $A$ can you find? My work: With a bit of elbow grease, I got $u(t) = \frac{1}{1-\omega^2}(\cos(\omega t)-\cos(t))$ I think I solved part c.). Since $\omega \geq 2$, we know that: $$|\frac{1}{1-\omega^2}| \leq 1, \hspace{10mm}|\cos(\omega t)-\cos(t)| \leq 2$$ Our worst case is $\omega = 2$ and $t = \pi + 2\pi n, n \in N$, so we get: $$|u(\pi)|=|-\frac{1}{3}(\cos(2\pi)-\cos(\pi))|=\frac{2}{3}$$ Take $A = \frac{2}{3}<1$ for all $t$. We see that as $\omega \to \infty$, we get an arbitrarily small $A$. Now part b.) is where I am a bit confused. Doesn't finding our value $A$ prove that $|u(t)|\leq 1$ for all $t$? Or does it need to be more rigorous?

A better way is to observe that $\omega \ge 2 \implies |1-\omega^2| \ge 3$, therefore

$$ |u(t)| = \frac{|\cos(\omega t) -\cos(t)|}{|1-\omega^2|} \le \frac23 $$