Prove or Disprove that there is an integer $n$ such that $n^4 + 1 < 2n^2$ I am a beginner in writing proofs and want to show that an integer $n$ such that the title is true does not exist. I went about it this way and am wondering if this is even considered a valid proof, and if not, some assistance.
First, let $n = 0$
$0^4 + 1< 2(0)^2$
$1 < 0$. False, $1 > 0$, therefore, at $n = 0$, $ n^2 > 2n^2$
Let us simplify $n^4 + 1 < 2n^2$
$= n^4 < 2n^2 -1$
$= (n^2)(n^2) < 2n^2-1$
$= n^2 < \frac{(2n^2-1)}{n^2}$
We know that the $\lim_{n\to\infty} n^2 = \infty$ and $\lim_{n\to\infty}\frac{(2n^2-1)}{n^2} \approx 2$
So we will use candidate values $n = 1$ and $n = 2$
$ (1)^2 < \frac{(2(1)^2-1)}{(1)^2}$
$=  1 < 1/2$
False, $1 > 1/2$, therefore, at $n = 1$, $ n^2 > \frac{(2(n)^2-1)}{(n)^2}$
Then,
$ (2)^2 < \frac{(2(2)^2-1)}{(2)^2}$
$=  4 < 7/4$
False, $4 > 7/4$, therefore, at $n = 2$, $ n^2 > \frac{(2(n)^2-1)}{(n)^2}$
We have already verified that at $n = 0$ and $n = 1$, $ n^2 > \frac{(2(n)^2-1)}{(n)^2}$. Because at  $n = 2$, the value of $n^2$ is already greater than $\lim_{n\to\infty}\frac{(2n^2-1)}{n^2} = 2$, and $\lim_{n\to\infty} n^2 = \infty$, $n^2$ will continue to grow to $\infty$ while $\frac{(2(n)^2-1)}{(n)^2}$ approches $2$. Thus we have shown that there is no integer $n$ such that $n^4 + 1 < 2n^2$
 A: IMHO, it's a (mostly) valid proof, but it's a bit sloppy. It does more work than it needs to, and neglects mentioning the case of $n<0$.
Also, good proofs are economical: they achieve the task with the minimal mathematical machinery required. Eg, you really don't need that stuff about limits. (BTW, $\lim_{n\to\infty}\frac{(2n^2-1)}{n^2} = 2$. The limit is exact, it's not an approximation).
The comments show a couple of very succinct ways to prove your statement. However, it's not expected that your proof is the smallest possible proof. The history of mathematics is full of examples of proofs by good mathematicians that were later superceded by new proofs that were more compact, elegant, or more general. The main thing is to write a proof in such a way that your readers will be able to follow your train of thought, and verify that your claims are legitimate. You'll get better at that with practice, and by reading good proofs written by others. In that respect, the art of proof writing is similar to that of writing computer programs.
