$M_n=\max\{S_k : k= 1,...n\}$
where $S_n$ is a simple random walk, $S_n = \sum_{i=0}^n X_i$ and $X_i$ is a Bernoulli random variable where $P(X = 1) = P(X = -1) = 0.5$
Prove that $P(M_n \geq r \bigcap S_n = v) = P(S_n = 2r - v)$.
I know that this requires using the reflection principle but I'm not sure where to go from here. We talked about this in class but I didn't understand the concept so I wanted to see if someone could clarify.