Introduction
In a panmictic population, the probability of fixation of an allele at a neutral locus is equal to its frequency at that time. I will refer to this probability of fixation as calculated at time $t$ as $P_{fix,t}$.
If $p_t$ is the frequency of the allele $A_1$ at time $t$, then the probability of the allele $A_1$ to reach fixation (rather than disappearing) is $P_{fix,t}=p_t$. Typically, the generation when the mutation occurred, the probability of the new allele to get fixed is $P_{fix,0}=p_0=\frac{1}{2N}$, where $N$ is the population size.
Question
This simple and classic result makes very good intuitive sense to me. However, I would fail to provide a mathematical proof.
Can you please demonstrate that $P_{fix,t}=p_t$?