Is it true $\forall\ n\ \exists\ p (p^2 \leq n < (p + 1)^2$) where the domain of the quantifiers is $\mathbb{N}$?

Question

Is it true $\forall\ n\ \exists\ p ( p^2 \leq n < (p + 1)^2$) where the domain of the quantifiers is $\mathbb{N}$?

I think this is true.

How to prove?

Answer 1

For given $n$, the set $$\tag1\{\,k\in\Bbb N\mid n<(k+1)^2\,\} $$ is non-empty because it contains $n$, for example: $$ (n+1)^2=n+(n^2+n+1)\ge n+1>n.$$ Hence $(1)$ has a minimal element $p$. If $p=\min\Bbb N$ (i.e., $p=0$ or $p=1$, depending on how you define $\Bbb N$), then $p^2=\min \Bbb N\le n$. If $p>\min \Bbb N$, then $p=k+1$ for some $k\in\Bbb N$, where by minimality of $p$, we have $n\not<(k+1)^2$, i.e., again $n\ge p^2$.

Answer 2

Hint

What about, $n=2$ or $n=3$ ? $ \ \ \ \ \ \ \ \ \ $