Why is the probability of a random walk reaching 1 (in n steps) squared greater than the probability of it reaching 2 (in n steps)?

Question

Let $S_n$ be a simple random walk. i.e.

$$ S_n = \sum_{t=1}^n X_t, $$ where ${X_t}$ are i.i.d random variables with

$$ X_t = \begin{cases} +1, & \textrm{w/ probability } p \\ -1, & \textrm{w/ probability } q=1-p \end{cases} $$

Let $$P_k(n) = \mathbb{P}(\textrm{reach } x=k \textrm{ within the } n \textrm{ first steps})$$

Is there an easy way to prove $P_1(n)^2 \geq P_2(n)$? I tried to think of ways through induction or relating $P_1(n)^2$ to $P_2(2n)$. To no avail.

Answer 1

$$\begin {array}{} P_2(n) &=& \sum_{k=1}^{n-1}P_1 (n-k)( P_1(k)-P_1 (k-1)) \\&\leq& \sum_{k=1}^{n-1}P_1 (n)( P_1(k)-P_1 (k-1)) \\&\leq& P_1(n) \sum_{k=1}^{n-1} (P_1(k)-P_1 (k-1))= P_1 (n) P_1 (n-1) \\ &\leq& P_1 (n) P_1 (n)\end {array} $$

Where $( P_1(k)-P_1 (k-1))$ is the probability to reach position 1 in the $k$-th step (which is different from within $k$ steps). And then $( P_1(k)-P_1 (k-1))P_1 ({n-k})$ is then the probability to reach another step in the direction 1 after being the first time in position 1 at the $k $-th step.