In the proof of Theorem 48.5. in Munkres' Topology it is written : I can understand the intersection of closed sets is closed, but it is closed for a single pair $n,m$ in the first place? I mean, why "the set of those $x$ for which $d(f_n(x), f_m(x)) \le \varepsilon$ is closed in $X$ by conti...

I am tasked to compute the integral $$ \int_0^\infty \frac{\log(x)}{1-x^{8}}\,dx\, $$ using contour integration. I've seen some approaches to complex logarithms, but never with such a function in the denominator. I can't seem to figure out which contour fits this problem best. If I choose a semic...

Compute the integral using Residue theorem: $$\int_{-\infty}^\infty {\cos x\over a^2-x^2}\ dx\quad a>0.$$ My attempt: Choose the contour $$\Gamma = \{Re^{i\theta}:0\leq\theta\leq\pi\}\cup [-R,-a-\epsilon]\cup\{-a+\epsilon e^{i\theta}:0\leq\theta\leq\pi\}\cup[-a+\epsilon,a-\epsilon]\cup\{a+\epsi...

Let $r,s$ be positive integers, and let $A=(0,0)$, $B=(1,0)$, $C=(r,s)$, and $D=(r+1,s)$. Find the coordinates of a point $P$ in terms of $r,s$, such that if $E$ is any point on $AB$, and $F$ is the unique point other than $E$ that lies on the circumcircles of triangles $BCE$ and $ADE$, then $P$ ...