An inequality in martingale

Question

Suppose $X_n$ is a supermartingale,for $\lambda>0$ ,we have the following inequality: $$\lambda\mathbb{P}(\inf_{n\leq k}X_n\leq-\lambda)\leq\int_{[\inf_{n\leq k }X_n\leq -\lambda]}(-X_k) \mathbb{dP}$$

I have tried to prove:

Let $T=\inf\{n:X_n \leq-\lambda\}\wedge k$, so $T$ is a bounded stopping time.By OST:

$$\mathbb{E}[-X_0]\leq\mathbb{E}[-X_T]=\int_{[\inf_{n\leq k }X_n\leq -\lambda]}-X_T \mathbb{dP}+\int_{[\inf_{n\leq k }X_n> -\lambda]}-X_T \mathbb{dP}$$ $$\geq\lambda\mathbb{P}(\inf_{n\leq k }X_n\leq -\lambda)+\int_{[\inf_{n\leq k }X_n> -\lambda]}-X_k \mathbb{dP}$$

Then I don't know how to do next since the two inequalities are not consistent.

Answer 1

The proof of Doob’s martingale maximal inequalities applies to the submartingale $−X$, see for example this one.