I'm studying martingales with the book Probability - Theory and Examples by R. Durrett and while keeping track of the dual results about supermartingales when something about submartingales is proven (by pluging $-X_n$ into the result when $X_n$ is a supermartingale) and vice versa, i ran into some problems with understanding what's happening in the supermartingale case for Doob's Inequality. Let $a<b$, the Upcrossing Inequality is stated for submartingales as follows
If $X_m$, $m\geq0$ is a submartingale, then $$(b-a)\mathbb{E}U_n\leq\mathbb{E}(X_n-a)^+-\mathbb{E}(X_0-a)^+\quad\quad (1)$$
where $N_0=-1$, $N_{2k-1}:= \inf\{m>N_{2k-2}: X_m\leq a\} $, $N_{2k}:=\inf\{m>N_{2k-1}: X_m\geq b\} $ and $$U_n:=\sup\{k:N_{2k}\leq n\}.$$
Applying this to the supermartingale $(Y_m)_m$ gives
$$(b-a)\mathbb{E}U_n\leq\mathbb{E}(Y_n+a)^--\mathbb{E}(Y_0+a)^-$$ where $N_{2k-1}:= \inf\{m>N_{2k-2}: Y_m\geq -a\} $, $N_{2k}:=\inf\{m>N_{2k-1}: Y_m\leq -b\} $. This can be interpreted as an upper bound for the number of downcrossings $D_n:=U_n$ on the reflected interval $[-b,-a]$. As $a<b$ are arbitrary constants, we can go back to the original interval by setting $-b:=a$, $-a:=b$. We get
$$(b-a)\mathbb{E}D_n\leq\mathbb{E}(Y_n-b)^--\mathbb{E}(Y_0-b)^-,$$ where $N_{2k-1}:= \inf\{m>N_{2k-2}: Y_m\geq b\} $, $N_{2k}:=\inf\{m>N_{2k-1}: Y_m\leq a\} $. (Note that the Stopping Times flipped their roles with respect to the submartingale case.)
Now observe that the number of upcrossings $U'_n$ in the last situation is $U'_n=D_n$ if there is a last upcross after the time $N_{2D_n}$ and before $n$ (that is, $N_{2D_n+1}\leq n$), and $U'_n=D_n-1$, otherwise. In any case, $$(b-a)\mathbb{E}U'_n\leq\mathbb{E}(Y_n-b)^--\mathbb{E}(Y_0-b)^-. \quad\quad(2)$$ The only differences between $U'_n$ and $U_n$ as in $(1)$ is that in $U'_n$ we are starting above $b$, waiting for a full downcross before starting the counter and we are not counting the last upcross (if there is any). So they are both bounded or both unbounded.
I am also aware of the following Upcrossing inequality for supermartingales, which can be proved in an even easier fashion than the proof Durrett provides for $(1)$
If $(Y_m)_m$ is a supermartingale, then $$(b-a)\mathbb{E}U_n\leq \mathbb{E}(Y_n-a)^-\quad\quad(3)$$
Question:
Are the inequalities $(1)$ and $(3)$ equivalent? If yes, could we recover $(3)$ from $(2)$?