is this finger pulse oximeter FDA approved?

yes, of course. the oximeter has a FDA

Questions & Answers: Pulse Oximeter Fingertip (Oximetro), Blood Oxygen Saturation Monitor, Heart Rate Monitor and SpO2 Levels, Portable Pulse Oximeter,USA Shipping

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Pulse Oximeter Fingertip (Oximetro), Blood Oxygen Saturation Monitor, Heart Rate Monitor and SpO2 Levels, Portable Pulse Oximeter,USA Shipping

Accurate & Fast reading - can be measured accurately in about 10 seconds or less.
Large LED display & Portable - Small size and the attached lanyard makes it easy to carry.
Low Power Consumption - The finger oximeter needs 2 AAA batteries,can be operated continuously for more than 30 hours. Also, the oximeter will power off automatically
You Will Get - 1* Pulse Oximeter, 1* User Manual and 1 year warranty.