Let $\{X_n\}_{n\geqslant 1}$ a sequence of i.i.d. Rademacher r.v., that is $\Pr [X_1=1]=\Pr [X_1=-1]=\frac1{2}$ and for $n,m\in \mathbb{Z}$ with $n<m$ set $X_0:\equiv x$ for some $x\in(n,m)\cap \mathbb{Z}$. Finally define the symmetric random walk $\{S^x_n\}_{n\in\mathbb{N}}$ given by $S^x_n:=\sum_{k=0}^n X_k$.
Let $\tau _p:=\inf\{n\in \mathbb{N}: S^x_n=p\}$ and set $\tau := \tau_{n}\wedge \tau _{m}$, that is, $\tau $ is the escape time of the set $(n,m)$. Now I want to prove that $\{S^x_{\tau \wedge n}\}_{n\in\mathbb{N}}$ is an uniformly integrable martingale. First suppose that $\Pr [\tau =\infty ]=0$, then $\tau \wedge n \to\tau $ a.s. and consequently $S^x_{\tau \wedge n}\to S^x_{\tau }$ a.s. also, then by the (reversed) Fatou's lemma we find that $$ \begin{align*} \limsup_{n\to\infty}\mathbb{E}[\mathbf{1}_{\{\tau \wedge n>\lambda \}}|S^x_{\tau \wedge n}|]&\leqslant \mathbb{E}[\mathbf{1}_{\{\tau >\lambda \}}|S^x_{\tau }|]\\&=\sum_{k\geqslant \lceil \lambda \rceil }\mathbb{E}[\mathbf{1}_{\{\tau =k\}}|S^x_k|]\\ &\leqslant \max\{|n|,|m|\}\cdot \Pr [\tau >\lambda ]\xrightarrow{\lambda \to \infty }0\tag1 \end{align*} $$ Therefore to prove that $\{S^x_{\tau \wedge n}\}_{n\in\mathbb{N}}$ is uniformly integrable it is enough to show that $\tau $ is a.s. finite. Now note that $$ \Pr [\tau =\infty ]=\Pr [S_k^x\in(n,m),\forall k\in \mathbb{N} ]\leqslant \Pr \left[\sup_{k\in \mathbb{N}}|S^x_k|<\infty \right]=\Pr \left[\sup_{k\in \mathbb{N}}|S^0_k|<\infty \right]\tag2 $$ and observe that $\left\{\sup_{k\in \mathbb{N}}|S^0_k|<\infty \right\}$ is a tail event in the tail $\sigma $-field generated by the $X_k$, so by the Kolmogorov's zero-one law we have that $\Pr \left[\sup_{k\in \mathbb{N}}|S^0_k|<\infty \right]\in\{0,1\}$. Now observe that for $M\geqslant 0$ we have that $$ \begin{align*} \Pr \left[\sup_{k\in \mathbb{N}}|S_k^0|\geqslant M\right]&\geqslant \Pr \left[\exists k_0\in \mathbb{N},\forall j\in \{k_0,\ldots ,k_0+2M-1\}:X_j=X_{j+1}\right]\\ &=1-\Pr \left[\forall k\in \mathbb{N},\exists j_0\in \{k,\ldots ,k+2M-1\}:X_{j_0}\neq X_{j_0+1}\right]\\ &\geqslant 1-\Pr \left[\forall k\in \mathbb{N},\exists j_0\in \{k2M,\ldots ,(k+1)2M-1\}:X_{j_0}\neq X_{j_0+1}\right]\\ &=1\tag3 \end{align*} $$ where the last equality follows from the fact that for every set $\left\{\exists j_0\in \{k2M,\ldots ,(k+1)2M-1\}:X_{j_0}\neq X_{j_0+1}\right\}$ we have that $$ \begin{align*} &\Pr [\exists j_0\in \{k2M,\ldots ,(k+1)2M-1\}:X_{j_0}\neq X_{j_0+1}]\\ &\qquad =1-\Pr [\forall j\in \{k2M,\ldots ,(k+1)2M-1\}:X_{j}= X_{j+1}]\\ &\qquad =1-\left(\frac1{2}\right)^{2M}\tag4 \end{align*} $$ what imply that $$ \begin{align*} &\Pr \left[\forall k\in \mathbb{N},\exists j_0\in \{k2M,\ldots ,(k+1)2M-1\}:X_{j_0}\neq X_{j_0+1}\right]\\ &\qquad =\Pr \left[\bigcap_{k\geqslant 0}\Big\{\exists j_0\in \{k2M,\ldots ,(k+1)2M-1\}:X_{j_0}\neq X_{j_0+1}\Big\}\right]\\ &\qquad =\prod_{k\geqslant 0}\Pr \left[\exists j_0\in \{k2M,\ldots ,(k+1)2M-1\}:X_{j_0}\neq X_{j_0+1}\right]\\ &\qquad =\lim_{k\to \infty }\left(1-\left(\frac1{2}\right)^{2M}\right)^{k}=0\tag5 \end{align*} $$ Therefore, as (3) holds for every $M\geqslant 0$, it follows that $$ \Pr \left[\sup_{k\in \mathbb{N}}|S^0_k|<\infty \right]=\lim_{M\to \infty }\Pr \left[\sup_{k\in \mathbb{N}}|S^0_k|< M \right]=0\tag6 $$ and so $\Pr [\tau =\infty ]=0$, as desired.∎
I think that the above proof is correct, however I will be grateful to have some confirmation. Another question is: the work to show that $\Pr [\tau =\infty ]=0$ seems a bit involved, so I guess there must be a simplest way to show it, can someone show me one?