monotonicity for random walks Consider a simple symmetric random walk on the integers. Let $\mu(k,n)$ denote the probability of being at $k$ by step $n$. Say both $n,k$ are even. I'm interested in finding "soft" proofs (minimal computations, e.g. by coupling) of two facts.
First:
$$\mu(k,n) < \mu(0,n),$$
Second, for $k^2 \ll n$,
$$\frac{\mu(k,n) }{ \mu(0,n)} = 1 + O(k^2/n)$$
Of course, these are not too bad by direct computation, but I am interested in some specific variants of these inequalities where direct computation become quite cumbersome. Looking to hear other styles of arguing for random walks.

For the first inequality to hold, $n$ must be even, so I will assume that.
The inequality is much more general, holding for any symmetric walk:
Considering the location of the walk after $n/2$ steps, we obtain from Cauchy-Schwarz that
(all the summations are over all integers):
$$\mu(k,n)^2=\Bigl(\sum_x  \mu(x, n/2 ) \cdot \mu(k-x, n/2 ) \Bigr)^2 $$ $$\le
\sum_x  \mu(x, n/2 )^2 \cdot \sum_x \mu(k-x, n/2 )^2 $$ $$= 
\Bigl(\sum_x  \mu(x, n/2 )   \mu(-x, n/2 )  \Bigr)^2= \mu(0, n )^2 \,.$$
