\begin{array}{l}{\text { let } T \text { be a stopping time with respect to}\text { the filtration }\left\{F_{n}, n \geq 1\right\} \text { assume that}} \\ {\text { there exists } c>0 \text { and } N \geq 1 \text { such that }} \\ {\qquad \begin{array}{ll}{\forall n \geq 1, \,\,\,\,\,\,P\left(T \leq n+N / \mathcal{F}_{n}\right)>c, \quad almost \,\,surely} \\ {\text { show that for all } k \in \mathbb{N},} \,\,{\text{ that :}\,\,\, P(T \geq k N) \leq(1-c)^{k}}\end{array}}\end{array}
I honestly couldn't do much except this : $P\left(T > n+N \right)\leq 1-c, \,\,\,\,\, \forall n \geq 1$
so we take $n = (k-1)N$ and $k$ is an arbitrary integer which gives us : $P(T \geq k N) \leq1-c$ but I'm missing the exponent
any help will be greatly appreciated.
\begin{alignat*}{2} \mathbb P(T\ge 2N) & = \mathbb P(T\ge 2N\;|\;T\ge N).\mathbb P(T\ge N) \\ & \le \mathbb P(T\ge 2N\;|\;\mathcal F_N).\mathbb P(T\ge N),\quad\text{as}\quad \mathcal F_N \supset \{T\ge N\}\\ & \le (1-c)^2 \end{alignat*}