Let $n \in \mathbb{N}^*.$ $(X_k)_k$ is a submartingale, $\lambda>0,$ by considering, for $1 \leq k \leq n, E_k=\bigcap_{r=1}^{k-1}\{X_r >-\lambda\}\cap\{X_k \leq -\lambda\},$ prove that $$\lambda P( \min_{1 \leq k \leq n}X_k\leq -\lambda) \leq \int_{\{\min_{1 \leq k \leq n}X_k> -\lambda\}}X_ndP - E[X_0].$$
$\{\min_{1 \leq k \leq n}X_k \leq -\lambda\}=\bigcup_{k=1}^nE_k,$ also $E_k$ are disjoint so $P(\min_{1 \leq k \leq n}X_k\ \leq - \lambda)=\sum_{k=1}^nP(E_k)$ also $\int_{E_k}X_kdP \leq -\lambda P(E_k).$
Also $X_k \leq E[X_n |\mathcal{F}_k].$
Having a problem how to continue from here, especially how to use that $(X_k)_k$ is a submartingale ?
Let $A=\{ \min_{1 \leq k \leq n}X_k\leq -\lambda\}$, and $$N=\inf\{m: X_m\leq-\lambda\text{ or }m=n\},$$ then $X_N\leq-\lambda$ on $A$ and $X_N=X_n$ on $A^c$. Hence $E[X_N1_A]\leq-\lambda P(A)$ and $$E[X_n1_{A^c}]=E[X_N1_{A^c}]=E[X_N]-E[X_N1_A]\geq E[X_N]+\lambda P(A).$$
Since $N$ is a bounded stopping time with $P(N\leq n)=1$ and $X_n$ is a submartingale, we have $$E[X_0]\leq E[X_N]\leq E[X_n].$$ Therefore $$E[X_n1_{A^c}]\geq E[X_0]+\lambda P(A),$$ as desired.