I'm a little unsure if my approach here is correct, which is making it difficult to find the limit. Any and all help will be appreciated!

Let {X_n} be a primitive finite state Markov chain with transition matrix P and stationary distribution w. Define the process Y_n by:
Y_n = (X_{n-1}, X_n)

Show that {Y_n} is a Markov chain, and compute:
lim_{n\to\infty} P(Y_n = (i,j))

What I've done:
So I need to show that Y_{n+1} is only determined by Y_n, ie:
P(Y_{n+1} | Y_{n}, Y_{n-1}, ... , Y_0) = P(Y_{n+1} | Y_{n})

So, suppose Y_n = (a,b) a & b are just some numbers. Then:
P(Y_{n+1} = (i,j) | Y_{n} = (a,b))
Plugging in the definition of {Y_n}:
= P((X_{n}=i, X_{n+1}= j) | (X_{n-1} = a, X_n=b))
Because {X_n} are primitive Markov-Chains, can I say that
= P((X_{n}=i, X_{n+1}= j)
because X_{n+1} doesn't depend on X_{n-1}? This is basically where I'm stuck