1. **State the problem:** Find the infimum of the set $$\left\{ \frac{m+n}{mn} \mid m,n \in \mathbb{N} \right\}$$ where $m$ and $n$ are natural numbers. 2. **Rewrite the expression:** The expression can be written as $$\frac{m+n}{mn} = \frac{m}{mn} + \frac{n}{mn} = \frac{1}{n} + \frac{1}{m}$$. 3. **Analyze the expression:** Since $m,n \in \mathbb{N}$, the smallest values for $m$ and $n$ are 1. 4. **Evaluate at smallest values:** For $m=1$ and $n=1$, $$\frac{1}{1} + \frac{1}{1} = 2$$. 5. **Check behavior as $m,n$ grow:** As $m$ and $n$ increase, $\frac{1}{m}$ and $\frac{1}{n}$ approach 0, so $$\frac{1}{m} + \frac{1}{n} \to 0$$. 6. **Determine infimum:** The values are always positive and can get arbitrarily close to 0 but never reach 0 because $m,n$ are natural numbers (no zero). 7. **Conclusion:** The infimum of the set is $$0$$, but it is not attained by any pair $(m,n)$. **Final answer:** $$\inf \left\{ \frac{m+n}{mn} \mid m,n \in \mathbb{N} \right\} = 0$$.