Is the following inequality about probability true?$$\sum_{i=1}^{\infty}P(T\geq i)\leq \sum_{i=0}^{\infty}cP(T\geq ci)$$ Here $c>1$, an integer. $T$ is just a random variable taking non-negative integer values.
I have been thinking about this for over 3 hours. My progress so far is only that$$\sum_{i=1}^{\infty}P(T\geq i) \leq c\cdot\sum_{i=1}^{\infty}P(T\geq i)$$ since $c>1$, but then in fact $P(T\geq i)\geq P(T\geq ci)$ since $T\geq ci$ implies $T\geq i$, so it seems I cannot achieve the $\leq$ conclusion.
This is one step in a proof of the expected value of Gambler's Ruin game's stopping time being finite. But I was stuck here trying to understand that. Any help would be much appreciated. Thanks a lot.
$$ \sum_{i\ge 1} \mathsf{P}(T\ge i)\le \int_0^{\infty} \mathsf{P}(T\ge t)\,dt=c\int_0^{\infty} \mathsf{P}(T\ge ct)\,dt\le c\sum_{i\ge 0}\mathsf{P}(T\ge ci). $$