# Sequences : Limits and Convergence

This question is from Advanced Calculus II class, it is more like introduction to real analysis.

Let f_n: R -> R be the sequence of functions given by f_n(x) = x/ ( 1 + nx^2)

a). Prove that the sequence f_n converges uniformly to a function f. What is f?

b). Prove that for each x in R-{0} ( all real numbers but 0 not included), the sequence f'_n(x) converges to f'(x).

c). Compare f'(0) and limit f'_n(0) as n goes to infinity. ( Justify your answer please).

d). For every a < b, find the limit

Limit ( as n goes to infinity) of the integral from a to b f_n d(alpha), where the integral is taken with respect to any nondecreasing function alpha: [a,b] -> R.

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#### Solution Summary

Limits and convergence of sequences are investigated. The solution is detailed and well presented.