Help with 2 problems about compact/pointwise convergence from Munkres - Topology

Hells_Kitchen

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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