= Solution
Every <unitary matrix> is unitarily diagonalizable. Part (ii) therefore proves the stronger <matrix> statement $\phi(A(t))=I$, not merely a statement about its <eigenvalues>. The <kernel of a group homomorphism> is a <normal subgroup>, and it contains every upper unipotent. By part (iii) their <normal closure> is the entire <group>. Thus
$$
\boxed{\phi(g)=I\quad\text{for every }g\in SL_2(\mathbb R).}
$$
This proves the <triviality of finite-dimensional unitary representations of SL2R>. In fact the argument used only the algebraic homomorphism property and finite-dimensional unitarity; it did not need continuity.
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