This problem comes from actuarial exams. We consider a continuous surplus process $$ U(t) = ct - \sum_{k=1}^{N(t)} X_k,$$ where $N(t)$ is Poisson process such that $\mathbb{E}N(t)=\lambda t$, and $X_k$ are iid random variables such that $\mathbb{P}(X_1 \in [0,10])=1$ and $\mathbb{E}X_1 = 2$. We also know that $c> 2\lambda$. I am trying to prove the following inequality: $$ 1 \leq \mathbb{E}[U(\tau)|\tau<\infty] \leq 5,$$ where $\tau = \inf(t\geq0: U(\tau) < 0).$
I'm not sure where to start from. It is clear that a similar inequality holds for $0$ and $10$ instead of $1$ and $5$. Moreover, $$\mathbb{E}[U(\tau)|\tau<\infty] = (c-2\lambda) \mathbb{E}[\tau|\tau<\infty],$$ which can suggest that the right way to approach this problem may be by investigating conditional expectation of $\tau$.
This question feels like it has a tricky way to solve it, though. I would appreciate any help.