$T$ is a linear operator on $V$ and $m$ is a nonnegative integer such that $\ker T^m = \ker T^{m + 1}$. Then $\ker T^{m + k} = \ker T^m$ for all positive integers $k$.
Clearly $\ker T^{m + i} ⊂ \ker T^{m + i + 1}$ for all positive integers $i$. Now suppose $v ∈ \ker T^{m + i + 1}$. Then $T^{m + i + 1} v = T^{m + 1}(T^i v) = 0$ so $T^i v ∈ \ker T^{m + 1} = \ker T^m$. Thus $T^m (T^i v) = T^{m + i} v = 0$ so $v ∈ \ker T^{m + i}$. Therefore, $\ker T^{m + i + 1} = \ker T^{m + i}$ for all positive integers $i$ and the result follows.