<p><span style="color:#27ae60;"><span style="font-size:20px;">Solving the Problem</span></span></p>
<p>We're asked to find the lowest value of $n$ for which the inequality holds true.</p>
<p>Firstly, since the left-hand side is a fraction, let's also convert the right-hand side to a fraction. $0.001$ is equivalent to $\frac{1}{1000}$ or $\frac{1}{10^3}$.</p>
<p>So, our equation now looks like this:</p>
<p>$\frac{1}{2^n}<\frac{1}{10^3}$</p>
<p>Now, let's multiply both sides by $2^n$ and $10^3$ to get rid of the fractions:</p>
<p>$10^3<2^n$</p>
<p>Now, we just need to find a value of $n$, such that $2^n$ will be bigger than $1000$.</p>
<ul>
<li>Since the question asks for the least value, we can start with the smallest answer choice $10$, and try evaluating $2^10$: we can see that we get $1024$, which is bigger than $1000$.</li>
<li>If we reduce our $n$ value to $9$, $2^9=512$ is too small</li>
<li>So, the lowest suitable integer value of $n$ would be $10$</li>
</ul>
<p>So, our answer must be <span style="color:#27ae60;">A</span>.</p>