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Here is my attempt. $X_t$ is a general random walk is when we have a sequence of independent and identically distributed random variables $Y_1,Y_2,...$ such that $X_0=0,$ $X_1=Y_1$, $X_2=Y_1+Y_2$ and so on which then we have $X_t=X_{t-1}+Y_t$.

If we let $Y_i=a$ where $a$ is any constant, then we have $X_t=at$. Then $E(X_t)=aE(t)$ which we can see it depends on $t$ so the mean is not constant so it is not even weakly stationary.

Does this work?

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You are asked to prove that $\{X_n\}$ cannot be stationary, so considering a special case is not good enough. You have to assume that $Y_i$'s are not $0$ random variables. Suppose $\{X_n\}$ is stationary. Then the distribution of $X_n$ does not depend on $n$. In particular $Y_1$ has same distribution as $Y_1+Y_2$. But $Y_1$ and $Y_2$ are i.i.d. This implies $Y_1=Y_2=0$ almost surely. [There are many ways of proving this. One way is to use characteristic functions. I will give more details if needed].

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