I have a very silly question about an inequality.
Let $n_x \geq 1$, integer, intuitively there's a unique integer $N_x \geq 1$ such that
$$ 32N_x - 31 \leq n_x \leq 32N_x $$
However I don't know why I'm struggling to prove rigorously that the statement is true. My attempt was to define the function
$$ N_x = N_x(n_x) = \left\lfloor \frac{n_x+31}{32} \right\rfloor $$
By the monotonicity of the floor function I have
$$ N_x \leq \frac{n_x+31}{32} \Rightarrow 32N_x-31 \leq n_x $$
which gives me the lower bound on $n_x$ for the upper bound
$$ N_x > \frac{n_x + 31}{32} - 1 = \frac{n_x - 1}{32} \geq 0 \Rightarrow N_x \geq \frac{n_x}{32} \Rightarrow n_x \leq 32N_x $$
Assuming everything is correct I'm puzzled if $N_x \geq \frac{n_x}{32}$, then I don't actually like the proof I just gave since it involves the guessing of the function $N_x(n_x)$, while instead I was specifically interested only in the existence of a unique solution, I'm not sure also that I've proved that the solution I gave is the unique one. I proved that there's a solution but not that such solution is eventually the unique one.
Is there a better way to prove the uniqueness of the solution without passing through the function I defined?