I'm studying the lecture notes by Philippe Clément about Gradient Flows in Metric Spaces.
Now the following problem arises ($X$ is a Hilbert space):
Definition: Let $\phi:X \to (-\infty, \infty]$ be proper (that is it does not only attain $\infty$), lower semicontinuous and $\alpha$-convex for some $\alpha \in \mathbf R$, that is $\phi - \alpha e$ is convex where $e(x) = \frac12 |x|^2$. Define \begin{equation} \psi(y) := \begin{cases} \frac1{2h} |y - x|^2 + \phi(y), &\text{if $y \in D(\phi)$},\\ \infty &\text{otherwise}. \end{cases} \end{equation} this function has a global minimizer (is a lemma) which will be denoted by $J_h x$. Further let $h \in I_\alpha$, that is $I_\alpha = (0, \infty)$ if $\alpha \geq 0$ and $(0, |\alpha|^{-1})$ if $\alpha < 0$. Now we set $\phi_h(x) = \psi(J_h x).$
Now the claim is that $\phi_{h_1}(x) \leq \phi_{h_2}(x)$ for $0 < h_2 < h_1 \leq h_\alpha$ and $x \in X$ where $h_\alpha$ is $1$ for nonnegative $\alpha$ and $\frac1{2|\alpha|}$ for negative $\alpha$. Why is this true? I can't seem to figure out, but probably I'm missing something simple. They claim that it should follow from the information which I have given.