Is there anything in the Bolzano-Weierstrass proof (for points in $\mathbb{R}^k$) that is analogous to using equicontinuity for functions in Arzela-Ascoli? Does the statement: “points are always equicontinuous” make sense? I feel that I may be forcing this reinterpretation, but I also feel that there might be some connection between the topology of points and […]

Let $f_n:[0,1]\to \mathbb{R}$ be a sequence of continuously differentiable functions such that $$f_n(0)=0,\:\: |f_n'(x)|\leq 1, \text{for all }n\geq 1, x\in (0,1).$$ Suppose further that $f_n(.)$ is convergent to some function $f(.)$. Show that $f_n(.)$ converges to $f(.)$ uniformly. I tired to prove this problem, but I’m lost and I cannot find a correct approach. Firstly, […]

The statement I am trying to prove: Let $\{ f_n \}$ be a sequence of equicontinuous, real valued, uniformly bounded continuous functions on $\mathbb{R}$. Show that $\{ f_n \}$ has a convergent subsequence which converges uniformly on any bounded subset of $\mathbb{R}$ and pointwise on all of $\mathbb{R}$ to a continuous function. (Royden pg 210, […]

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