Simple Random Walk - Why are these two events the same?

Question

Let $S = (S_n)_{n \geq 1}$ be a simple random walk. We denote the hitting time of a point $b$ by $\tau_b = \min \{i \geq 1 : S_i \geq b\}$.

My text says that the events $\displaystyle\{\max_{k \leq n} S_k \geq b\}$ and $\{\tau_b\ \leq n\}$ are identical.

I'm having trouble reading the first set, which seems to be about the maximum value (the values on the $y$-axis of the simple random walk) of $S_k$, for $k \leq n$, but the second set seems to be about "time" (the values on the $x$-axis of the simple random walk). So how can the events be identical? Am I misunderstanding something here?

Answer 1

It turns out that what is trying to be said is that $\mathbb{P}\displaystyle\{\max_{k \leq n} S_k \geq b\}$ and $\mathbb{P}\{\tau_b\ \leq n\}$ are equal, which they indeed are.

The question otherwise makes no sense.

Answer 2

If $\tau_b\leqslant n$, then $S_j\geqslant b$ for some $1\leqslant j\leqslant n$, and hence $$\max_{1\leqslant k\leqslant n}S_k\geqslant S_j\geqslant b.$$ If $\max_{1\leqslant k\leqslant n}S_k\geqslant b$, then $\tau_b\leqslant \operatorname{argmax}_{1\leqslant k\leqslant n}S_k\leqslant n$.

It follows that the two sets are equal.