$X_n$ s.t $E[\sup _n |X _{n+1}-X_n|]< \infty$ and $X_0 =0$. Prove that a.s $\limsup X _n=-\liminf X _n =\infty$ or $\lim X _n$ exsits and finite

Question

$X_n$ s.t $E[\sup _n |X _{n+1}-X_n|]< \infty$ and $X_0 =0$
Prove that a.s $\limsup X _n=-\liminf X _n =\infty$ or $\lim X _n$ exsits and finite.
hint: you can prove ${Y _n}$ martingale then $\sup E[Y _n ^+]<\infty$ or $\inf E[ Y_n ^-]<\infty$

My solution did not use the hint so I want to make sure it is correct, and I would like to know how to prove the hint.

My Solution:
Denote $M=E[\sup |X _{n+1}-X _n|]<\infty$, $T=\inf [n:X _n <-K]$ we notice $X _{n\wedge T}$ is a martingale and $X _{n\wedge T}>-M-K$
If ${T=\infty}$ then $\lim X _n $ exsits and is not $-\infty $
Now we take $K\rightarrow \infty$ and get ${T=\infty}$ if ${\liminf X _n >\infty }$.
We do the same to the other side and get If ${T'=\infty}$ then $\lim X _n$ exsits and is not $\infty $

And so we get the claim.