Let's consider submartingale, so variable that satisfies :
$$E[X_{n+1}|\mathbb{F}_n] \ge X_n$$
Also let's assume that $\forall_{i, j}:E[X_i]=E[X_j]$
I want to prove that $X_n$ is martingale.
My work so far
To prove martingale, we only need to prove that X is supermartingal i.e. $$E[X_{n+1}|\mathbb{F}_n] \le X_n$$ From very first assumption we have that :
$$E[E[X_{n+1}|\mathbb{F}_n]] \ge E[X_n] \Leftrightarrow X_{n+1} \ge E[X_n]$$
Now from second assumption (equality of expectations) we have that $$X_n \ge E[X_n]$$
And now I stacked. Could you please give me a hand ?
Denote $Z_n:=\mathbb E[X_{n+1}\mid \mathcal F_n]$. You have that $$Z_n-X_n\geq 0.$$ Since $(X_n)$ has constante expectation, $$\mathbb E[Z_n-X_n]=\mathbb E[X_{n+1}]-\mathbb E[X_n]=0,$$ and thus $Z_n=X_n$ a.s. as wished.