Given $f:[0,\infty)\to \mathbb{R} $ a uniformly continuous function and $g:[0,\infty)\to \mathbb{R} $ a continuous function. If $$ \lim_{x \to \infty} g(x) - f(x) = 0$$ is g uniformly continuous over $[0, \infty)$? Thank you for your help!
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1What have you tried so far? Have you considered the case $f\equiv 0$? – Severin Schraven Mar 28 '24 at 16:33
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What is the argument $i$ in your limit? – Julius Mar 28 '24 at 16:35
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Thank you, that was meant to be an x – Ayelet Wasserman Mar 28 '24 at 16:42
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Hi, welcome to MSE. Is it meant to be "$x \to \infty$"? – Izaak van Dongen Mar 28 '24 at 16:43
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As Severin says, you should consider $f = 0$ first. This is discussed here. – Izaak van Dongen Mar 28 '24 at 17:01
