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The intermediate value theorem states that if $f:[\alpha ,\beta ]\to \mathbb{R}$ is continuous and $y$ lies between $f(\alpha )$ and $f(\beta )$, then some $c\in [\alpha ,\beta ]$ satisfies $f(c)=y$.

It is enough to prove the caseLetChoose $x_n\in E$ with $x_n\to c$. Continuity gives $f(c)\le y$. If $f(c)<y$, continuity would make $f(x)<y$ for some $x>c$, contradicting that $c$ is an upper bound of $E$. Thus $f(c)=y$. The case with reversed endpoint inequalities follows by replacing $f$ with $-f$.

The Flatness problem is that the observed cosmological density parameter is close to $\Omega =1$, although in an ordinary decelerating universe any departure from one grows with time. Extrapolation backward therefore requires implausibly precise early cancellation of the curvature term.

The cosmological perfect-fluid continuity equation isConsequentlyFor an expanding universe $H>0$. Ifthen $\rho a^2$ increases. The Friedmann equation impliesIt therefore decreases during this period, driving $\Omega$ toward one. By the Friedmann acceleration equation, the same condition gives $\ddot{a}>0$ when the cosmological constant is included in the effective fluid. This is the inflationary solution of the flatness problem.

For the scalar field, the slow-roll approximation meansThe equations reduce toTaking the positive-field branch gives $H\simeq m\varphi /\sqrt{3}$, and thereforeWith $\varphi (0)={\varphi }_i$,Now integrate $H=\dot{a}/a\simeq m\varphi /\sqrt{3}$:Thus the quadratic-potential slow-roll solution isFinally,so equivalently

An adiabatic invariant remains constant to leading order when a system parameter changes on a timescale much longer than one period. For a periodic one-degree-of-freedom system, the adiabatic invariance of the action supplies such an invariant.

The momentum magnitude between collisions is $p_0=\sqrt{2mE}$. A complete orbit crosses the gap once in each direction, so the unnormalized action used in the question isConsequentlybecause the speed is $\sqrt{2E/m}$ and the round-trip distance is $2L$.

When $L(t)$ varies adiabatically, conservation of the adiabatic particle between moving parallel walls action givesA moving wall does work at each collision: an approaching wall raises the particle's energy and a receding wall lowers it. Elasticity holds in the instantaneous rest frame of the wall.