Why are these two definitions of Martingales equivalent? I am recently reading books on Probability theory, in Durrett's book Probability: Theory and Examples, the definition is following:

If $X_{n}$ is sequence with
(1) $\mathbb{E}\left|X_{n}\right|<\infty$
(2) $X_{n} \text { is adapted to } \mathcal{F}_{n}$
(3) $\mathbb{E}\left(X_{n+1} | \mathcal{F}_{n}\right)=X_{n} \text { for all } n$
Then $X$ is said to be a martingale (with respect to $\mathcal{F}_n$ )

And, in Erhan Cinlar's book Probability and Stochastics, the definition is

A real-valued stochastic process $X=\left(X_{t}\right)_{t \in \mathbb{T}}$ ($\mathbb{T}$ is a subset of $\overline{R}$.) is called $\text {an } \mathcal{F} \text { -submartingale if } X \text { is adapted to } \mathcal{F}, \text { each } X_{t} \text { is integrable, and}$ $$\mathbb{E}_{s}\left(X_{t}-X_{s}\right) \geq 0$$
$\text { whenever } s<t$, $\text {It is called an } \mathcal{F} \text { -supermartingale if }-X \text { is an } \mathcal{F} \text {-submartingale}$ $\text { and an } \mathcal{F}\text{-martingale} $ is it's both $\mathcal{F} \text { -supermartingale}$ and $\mathcal{F} \text { -submartingale}$.

From first looks I don't see the connection immediately, can you please illuminate me on this?

From Cinlar's definition, if $X_t$ is a martingale, it is both a supermartingale and a submartingale.
The submartingale property implies $E_s[X_t - X_s] \ge 0$, and the supermartingale property implies $E_s[-(X_t - X_s)] \ge 0$, and thus $E_s[X_t - X_s] = 0$. Since $X_s \in \mathcal{F}_s$, this can be rewritten as $E_s[X_t] = X_s$. (Presumably, $E_s[\cdot]$ in Cinlar's notation is $E[\cdot \mid \mathcal{F}_s]$ in Durrett's notation?)
