$\int_{\mathbb R} \sup_{|y|>|x|} \frac{1}{(1+|y|)^{n}} dx < \infty$ for large $n$?

Question

It is true that $\int_{\mathbb R} \frac{1}{(1+|y|)^{n}} dy <\infty$ for large $n\in \mathbb N.$

My Question is: Can we expect $\int_{\mathbb R} \sup_{|y|>|x|} \frac{1}{(1+|y|)^{n}} dx < \infty$ for large $n$ again ?

Answer 1

Obviously, $$\int_{\mathbb R} \sup_{|y|>|x|} \frac{1}{(1+|y|)^{n}} dx = \int_{\mathbb R} \frac{1}{(1+|x|)^{n}}dx=\cdots$$

Answer 2

$$ \sup_{|y|>|x|} \frac{1}{(1+|y|)^{n}} = \frac{1}{(1+|x|)^{n}}$$

and $$\int_{\mathbb R} \frac{1}{(1+|x|)^{n}} dx = 2\int_0^\infty\frac{1}{(1+|x|)^{n}} dx \leq 2\left(\int_0^1 1 dx +\int_1^\infty\dfrac{1}{x^n}dx\right) $$