The 'direct proof' is a little 'efforts spending'. A confortable alternative is to read...

Möbius Inversion Formula -- from Wolfram MathWorld

...where the 'Moebius inversion pair formula' is illustrated...

(1)

Here is the 'Moebius function'. Of course the (1) can be apply 'from top to bottom' or 'from bottom to top'... using the last option You derive that...

(2)

Kind regards