<s>[INST] Integer within certain interval. I have to show that there is a unique integer within the interval $[a,b]$ where
$a = -\frac{3}{2} + \sqrt{\frac{9}{4} + 2(n+1)}$
and
$b = -\frac{1}{2} + \sqrt{\frac{1}{4} + 2(n+1)}$
as well as $n \in \mathbb{N}$.
Now the uniqueness is easy by showing that there can be one integer within this interval at maximum. This is done by calculating $(b-a)$ and finding that $(b-a)<1$.
But how do I prove that there indeed exists one?
 [/INST] Assume the contrary - $\nexists$ an integer lying in $[a,b]$. Then, $\exists\  m \in \mathbb{N}$ such that $[a,b] \subset [m,m+1]$.
So, 
\begin{align}
m &< a \\
\implies m+\frac{3}{2} &< \sqrt{\frac{9}{4} + 2(n+1)} \\
\implies m^2 + 3m &< 2n+ 2
\end{align}
and 
\begin{align}
b &< m+1 \\
\implies \sqrt{\frac{1}{4} + 2(n+1)} &< m+\frac{3}{2} \\
\implies 2n +2 &< m^2 + 3m + 2 \\
\implies 2n &<
m^2 + 3m\end{align}
Hence, we must have $m(m+3) = 2n+1$. But, $m(m+3)$ is an even number, while $2n+1$ is odd. 
This is a contradiction, and so the result is proved.
</s>