Solved – Prove that a simple random walk is a martingale

Note that $a$ has a mean of 0.

My approach:

$$E[X_{t+1}mid X_1 + dots+X_{t-1}]$$
$$=E[X_{t-1}+2amid X_1 + dots+X_{t-1}]$$
$$=E[X_{t-1}mid X_1 + dots+X_{t-1}]+E[2amid X_1 + dots+X_{t-1}]$$
$$=E[X_{t-1}mid X_1 + dots+X_{t-1}]+0$$
$$=E[X_{t-1}mid X_1 + dots+X_{t-1}]$$
Am I doing something wrong here? shouldn't the end product be $X_t$?

begin{align} E[X_{t+1} mid X_1, ldots, X_t] &= E[X_t + a_{t+1} mid X_1, ldots, X_t] \ &= X_t + E[a_{t+1} mid X_1, ldots, X_t] \ &= X_t end{align}

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