This is a question I can't really figure out with the hint that is given.
Consider a simple random walk, i.e., we have iid sequence $U_1, U_2,\dots$ with $\mathbb P(U_i=-1)=\mathbb P(U_i=1)=\frac{1}{2}$.
Define $X_0=0$ and for $n\geq 0$: $X_n=\sum_{i=1}^nU_i$.
Define $T_k=\inf\{n\geq 0\mid X_n=k \}$.
Show that for $k\neq 0$ and $n\geq 2$ that $\mathbb{P}(T_k>n)\geq \frac{2}{n+3}$ and conclude that $\mathbb{E}(T_k)=+\infty$.
The hint that is given is "Imagine $k>0$, how many steps does it take the walker at least to walk from $0$ to $-\frac{1}{2}n$ and then to $k$?"
That seems quite obvious, namely 2 times $\frac{1}{2}n$ steps to get to $\frac{1}{2}n$ and back to $0$. And then $k$ steps to position $k$, so $n+k$ steps. However, I don't see how that helps answering the question.