Internal Rate of Return vs Discount Rate

The IRR will be greater than $5\%$. The IRR is the discount rate $k_0$ at which one breaks even, i.e., $\operatorname{NPV}(k_0) = 0$. Assume that the IRR is unique. If $\operatorname{NPV}(k) > 0$, that is, the net present value using a discount rate of $k$ is positive, then the discount rate $k$ must be lesser (better) than the discount rate at which one breaks even; which is to say that $k < k_0$. In this context, $k_0$ is the IRR and $k = 0.05$. Here is a rigorous proof of this fact: Suppose that the IRR is unique. The function $\operatorname{NPV} \colon [0, \infty) \to \R$ is a continuous real valued function and thus the Intermediate Value theorem applies. One one hand, we have that $\operatorname{NPV}(0.05) = 18,000$. On the other, one calculates $$\lim_{k \to \infty} \operatorname{NPV}(k) = \lim_{k \to \infty} \sum_{n=0}^{\infty} \frac{c_n}{(1+k)^n} = c_0 = -20,000 < 0.$$ Thus, by the Intermediate Value Theorem, there must be some value $k_0 \in (0.05, \infty)$ such that $\operatorname{NPV}(k_0) = 0$; by uniqueness this is the IRR.