A basic estimate for Sobolev spaces Here is a statement that I came upon whilst studying Sobolev spaces, which I cannot quite fill in the gaps:
If $s>t>u$ then we can estimate:
\begin{equation}
(1 + |\xi|)^{2t} \leq \varepsilon (1 + |\xi|)^{2s} + C(\varepsilon)(1 + |\xi|)^{2u}
\end{equation} 
for any $\varepsilon > 0$
(here $\xi \in \mathbb{R}^n$ and $C(\varepsilon)$ is a constant, dependent on $\varepsilon$).
How can I show this? Many thanks for hints!

Let $f(x)=(1+x)^{2(t-u)}-\varepsilon(1+x)^{2(s-u)}$. We have $f(0)=1-\varepsilon$ and since $s-u>t-u$ and $\varepsilon>0$, we know $f(x)\to-\infty$ as $x\to\infty$. Hence $C(\varepsilon):=\displaystyle\sup_{0\leq x&lt\infty}f(x)&lt\infty$.
