Let $\sqrt{n}S^{n-1}$ be the sphere of radius $\sqrt{n}$ and consider the hemisphere given as $H = {\vec{x} \in \sqrt{n}S^{n-1}: x_1 \le 0}$.
Now I want to show that $$S = \{\vec{x} \in \sqrt{n}S^{n-1}| x_1 \le \frac{t}{\sqrt{2}}\} \subset H_t =\{\vec{x} \in \sqrt{n}S^{n-1}| \exists \vec{y} \in H : ||\vec{x}-\vec{y}||_2 \le t\}.$$
I tried something but I can see how to generalize it to more $\mathbb{R}^n$ for $n \ge 3$.
Assume $\vec{x} \in S$.
Case 1: $x_1 \le 0.$ In this case $\vec{x} \in H$ and we are done.
Case 2: $0 < x_1 \le \frac{t}{\sqrt{2}}.$ Take $\vec{y} = (0,sgn(x_2))$. clearly $\vec{y} \in H$ and $\|\vec{x}-\vec{y}\|_2 \le t.$
Any hints will be appreciated.