**Definitions:**

Let $xi_n$ be a symmetric random walk, i.e.,

$$

xi_n=eta_1+eta_2+dots+eta_n,

$$

where ${eta_n}$ is a sequence of i.i.d. random variables such that

$$

P{eta_n=1}=P{eta_n=-1}=frac{1}{2}.

$$

Furthermore, we define the first hitting time to be $$tau=minleft{n:|xi_n|=Kright},$$

where $K$ is a positive integer.

I was reading a book on stochastic processes and here we want to show that $tau<infty$ a.s. The book proves this as follows

We want to show that $P{tau=infty}=0.$ To this end we shall estimate $P{tau>2Kn}.$ Notice that $$P{tau>2Kn}le left(1-frac{1}{2^{2K}}right)^nto0$$ as $ntoinfty.$ Thus, we have begin{align}

P{tau=infty}&=bigcap_{n=1}^infty P{tau>2Kn} \ &=lim_{ntoinfty} P{tau>2Kn}=0.

end{align}

After spending so many time, I could not figure out how to get the inequality $$P{tau>2Kn}le left(1-frac{1}{2^{2K}}right)^n$$ in the first line of the proof. Can someone help me understand why this inequality holds?

Many thanks in advance.