# How to prove this inequality? It is true or false? [closed]

I'm trying to prove this statement:

Let $$\alpha, t >0$$, then there exists a positive constant $$c$$ such that $$\left( 1+\frac{t}{2}\right)^{-\alpha} \leq c \left( 1+t\right)^{-\alpha}.$$ Is this true? How could i proceed?

## closed as off-topic by John Omielan, YiFan, Thomas Shelby, Lord Shark the Unknown, max_zornApr 19 at 4:54

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• Are $\alpha,t>0$ fixed? – Dbchatto67 Apr 18 at 16:45
• Yes they are fixed. – C. Bishop Apr 18 at 16:47
• Then you can simply use Archimedean property of real numbers to prove that inequality. – Dbchatto67 Apr 18 at 16:49

• Archimidean property: Given any positive $x$ and $y$ in A, there is an integer $n$ such that $nx> y.$ In this case: $\left( 1+\frac{t}{2}\right)^{-a}$ and $\left( 1+t\right)^{-a}$ are fixed, then exists an integer $c$ such that $$\left( 1+\frac{t}{2}\right)^{-a} \leq c\left(1 +t\right)^{-a}.$$ Isn't it? – C. Bishop Apr 18 at 16:59