Solution
= Solution
For $x\geq0$, the <geometric distribution> tail gives
$$
P(T_k>x)=P(S_1>kx)
=\left(1-\frac1k\right)^{\lfloor kx\rfloor}
\longrightarrow e^{-x}.
$$
For $x<0$, $P(T_k\leq x)=0$. The limiting <cumulative distribution function> is therefore $0$ for negative $x$ and $1-e^{-x}$ for nonnegative $x$, and it is continuous everywhere. This proves
$$
\boxed{T_k\xrightarrow{d}\operatorname{Exp}(1),}
$$
with rate parameter $1$ and mean $1$.