I am trying to show:  If $(x_n)$ converges to $a&gt;0$ and $x_n&gt;0, y_n \geq 0$ then $\lim \sup (x_n y_n) = a\lim \sup y_n$. But have not been able to make much progress.  I can show that if $p$ is a limit point of $a(y_n)$ then it is a limit point of $(x_n y_n)$ but I cannot show the converse of this, which would complete the proof. Any hints?
I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n  $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n  ) = \limsup a_n \limsup b_n  $$ Help please, it´s not a homework I want to learn.
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Suppose that f is continuous on $(a,b)$ and $\lim_{x\to a^+}  f(x) = \lim_{x\to b^-} f(x) = \infty$. Prove that $f$ has a minimum on all of $(a,b)$.  I was thinking of applying the extreme value theorem, but I realized that it applies only to closed intervals. So I was thinking of defining a closed interval that is in $(a,b)$, and using the definition of one sided limits to go about the problem, but I don't know how to proceed. Help please? How do I use RelativeSource with WPF bindings and what are the different use-cases? Flight Search from anywhere to a specific destination Many flight search engines allow you to pick a departure airport and search for flights to anywhere. E.g.  gives you a nice map with prices next to destinations for a set of dates and an origin you selected. We covered this question widely in  here on our site.  I want to have the reverse, i.e. I know my dates and my destination (for a one-way flight) and I want to know from where it is cheapest to fly there. 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The  characterizes a collision in one dimension by relating the initial and final speeds of the particles involved,  $$C_R = -\frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$$   In a 2D collision, the velocity can be split into components parallel and perpendicular to the "plane of collision" (the plane tangent to the two objects' surfaces at their contact point), and the equation above applies to the perpendicular components of the velocities only. One could write  $$\begin{pmatrix}v_{2f\shortparallel} - v_{1f\shortparallel} \\ v_{2f\perp} - v_{1f\perp}\end{pmatrix} = \begin{pmatrix}1 &amp; 0 \\ 0 &amp; -C_R\end{pmatrix}\begin{pmatrix}v_{2i\shortparallel} - v_{1i\shortparallel} \\ v_{2i\perp} - v_{1i\perp}\end{pmatrix}$$   My question is, is it useful (i.e. does it produce a more accurate description of realistic 2D collisions) to generalize that matrix? Perhaps by allowing the top left element to be unequal to 1, or allowing nonzero off-diagonal elements? 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