8 problems.
- Word definitions of surjective, not injective, uniformly continuous, and not uniformly continuous
- The sequence \(a_{n+1}=\frac{a_n}{3}+5\) with \(a_1=1\) is increasing, bounded by \(10\), and converges to \(\frac{15}{2}\)
- \(\lim_{x\to 0}f(x)=0\) and \(g\) bounded imply \(f(x)g(x)\to 0\), with a counterexample if \(g\) is unbounded
- \(\{v_1+v_2,\, v_2-v_3,\, v_2+2v_3\}\) is a basis whenever \(\{v_1,v_2,v_3\}\) is
- Green’s theorem for \(\int_C xe^x\,dx+(ye^y+x^2)\,dy\) over the upper half of \((x-1)^2+y^2=1\)
- Spectral theorem diagonalization with eigenvalues \(6\) and \(1\), and the minimum of \(\langle Ax,x\rangle\) on the unit circle
- Equivalence relation \(x\sim y\) iff \(x^2-4x=y^2-4y\)
- Total differentiability at the origin when \(f_x\) is continuous