Let $t\in (0, 2)$ and let $v_n(t) = \sqrt{(t-1)^2+1/n}$. I am pretty sure that $v_n(t)\to v(t)$ uniformly, with $v(t) = |t-1|$.
Could someone please help me in proving this formally?
Thank you in advance!
2026-05-13 19:43:25.1778701405
Prove formally that $v_n(t)\to v(t)$ uniformly
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By the monotonicity and the subadditivity of the square root you have that $$ \vert t-1 \vert \leq v_n(t)\leq \vert t-1\vert+\frac{1}{n}$$; consequently $$ \left\| v_n(t) - \vert t-1 \vert \, \right\| \leq \frac{1}{n}, $$ and you conclude because the estimate does not depend by $t.$