help with 2 problems about compact/pointwise convergence from Munkres  Topologyby Hells_Kitchen Tags: compact or pointwise, convergence, munkres, topology 

#1
Dec709, 10:04 PM

P: 62

Hi everyone,
I am stuck with 2 problems from Munkres' book and I would appreciate if someone helped me solve them. Thank you in advance. Here they are: 1. Consider the sequence of continuous functions fn : ℝ > ℝ defined by fn(x) = x/n . In which of the following three topologies does this sequence converge: uniform, compact convergence, pointwise convergence? Answer the same question for the sequence given as: fn(x)= 1 / [n^3 * (x  1/n)^2 + 1] 2. Let (Y,d) be a metric space; let fn: X > Y be a sequence of continuous functions; let f: X > Y be a function (not necessarily continuous). Suppose that fn converges to f in the topology of pointwise convergence. Show that if {fn} is equicontinuous then f is continuous and fn converges to f in the topology of compact convergence. 


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