# A two-liner

Is there a ring $A$ such that there are two different ring injections $\Bbb{R}\to A$?

# Finite codimension

Let $E,F$ be Banach (more generally Fréchet) spaces and let $T,S:E\to F$ be continuous linear maps. Suppose that $T$ is surjective and $S$ is compact. Prove that $T+S$ has closed image and $\dim \mathrm{im}(T+S)<\infty$.

# Still expecting my fix

What’s the expected number of fixed points of a uniformly distributed random permutation of $n$ elements?

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# What goes up must come down

Let $f \in C^1(\mathbb{R})$ such that $f,f' \in L^1(\mathbb{R})$. Prove that $\int_\mathbb{R}f'=0$.