For how many positive integers $m$ does there exist at least one positive integer $n$ such that mn < $m$ + n?

Step 1 (Analyze the Inequality): We are looking for positive integers $m$ such that there exists at least one positive integer $n$ satisfying:

Step 2 (Test a Specific Value for n): Let's test the smallest positive integer for $n$, which is $n=1$. Substituting $n=1$ into the inequality:

Step 3 (Evaluate the Resulting Inequality): The inequality $m<m+1$ is a fundamental arithmetic fact that is strictly true for all real numbers, and thus true for all positive integers $m$.

Step 4 (Conclusion): Since choosing $n=1$ satisfies the inequality for every positive integer $m$, there exists at least one positive integer $n$ (namely $n=1$) for every single positive integer $m$. Therefore, there are infinitely many such positive integers $m$.