If your arms are cold then your hands wont warm up either.
Not true really- hat and gloves will keep you warm. Used to live in the Alps and ran in vest shirts but with gloves and hat and was always ok
Running and cycling aren't the same though. Wind chill for cycling is significantly greater and it effects your arms more as they are also out there unshielded. If the blood in your arms is cold your fingers will be too.
@Grande-ginge started
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If your arms are cold then your hands wont warm up either.