Why is $\mathbb{P}(f_{j+1}=m, f_i=l | f_0=k) \leq \mathbb{P}(f_{j+1}=m | f_0 =k)$?

Question

Why is $\mathbb{P}(f_{j+1}=m, f_i=l | f_0=k) \leq \mathbb{P}(f_{j+1}=m | f_0 =k)$?

Where $i,j \geq 0$ and nothing is assumed by the order of probabilities.

And where $(f_i)_i$ is a Markov chain that's homogenous.

Answer 1

$P(A\cap B|C)=\frac {P(A\cap B\cap C)} {P(C)}\leq \frac {P(A\cap C)} {P(C)}=P(A|C)$ always. Markov property is not required.