Let $k\gt0$ and consider the real sequence $(u_n)_{n\geqslant1}$. Moreover, let \begin{equation} T_k u_n :=\begin{cases} u_n &\hbox{ if } \vert u_n\vert\leqslant k\\[5pt] k\frac{u_n}{\vert u_n\vert} &\hbox{ if } \vert u_n\vert > k \end{cases} \end{equation} and define $v_{k, n}:= T_k u_n -u$.
How could I prove that $$ \vert\varphi(v_{k, n})\vert\leqslant\varphi(2k) \quad \mbox{ and }\quad 0 <\varphi^{\prime}(v_{k,n})\leqslant\varphi^{\prime}(2k),$$ where $\varphi(t)$ denotes the real map $\phi(t) = t e^{\eta t^2}$, with $\eta\gt 0$?
May I ask for help? Thank you in advance!