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## Homework Statement

Define

[itex]f_n : \mathbb{R} \rightarrow \mathbb{R} [/itex] by

[itex]f_n(x) = \left( x^2 + \dfrac{1}{n} \right)^{\frac{1}{2}}[/itex]

Show that [itex]f_n(x) \rightarrow |x|[/itex] converges uniformly on compact subsets of [itex]\mathbb{R}[/itex]

Show that the convergence is uniform in all of [itex]\mathbb{R}[/itex]

## The Attempt at a Solution

Not quite good at these epsilon proofs, not sure if it needs to go that far but by the root law we have that

[itex]\lim_{ n\rightarrow \infty} \sqrt{x^2 + \dfrac{1}{n}} = \sqrt{x^2 + \lim_{n \rightarrow \infty} \dfrac{1}{n}} = \sqrt{x^2} = |x|[/itex]. Now this shows that [itex]f_n [/itex] converges uniformly in some subset of [itex]\mathbb{R} [/itex] correct?