Stochastic processes, White noise.
Solution 1:
$$Cov(X_n,X_{n-1})$$ $$=Cov ((X_n-X_{n-1})+X_{n-1}, X_{n-1})$$ $$=Cov ((X_n-X_{n-1}), X_{n-1})+Cov (X_{n-1}, X_{n-1})$$ $$=0+Cov (X_{n-1}, X_{n-1}).$$ This is true for any process with independent increments and finite second moment, in particular for the Poisson process.