Solution
= Solution
For every <vector> $z\in\mathbb R^n$, splitting the rows gives
$$
\|A_1z\|_2^2+\|A_2z\|_2^2=\|Az\|_2^2=z^TA^TAz=\|z\|_2^2.
$$
Both summands are nonnegative. Taking the supremum over <unit vectors> in the definition of the <spectral norm> proves
$$
\boxed{\|A_1\|_2\leq1,\qquad \|A_2\|_2\leq1}.
$$