Most Soil Mechanics and Foundations text and reference books (such as NAVFAC DM 7.01 and Verruijt) state the equations for Boussinesq’s point load problem without proof. For those who are interested in how these equations are developed, below is the derivation, taken from Manual of the Theory of Elasticity, by V.G. Rekach, where more detail is given along with the notation, which is different from what we have in the U.S.. The derivation from Rekach is given below.
Derivation of Boussinesq’s Stress and Strain Equations for a Point Load on a Semi-Infinte Space (like soil)
vulcanhammer.net 
Thank you. I was trying to find the proof I only found old french books which didn’t make much sense I will try to read this one, even though as you said it has different notation. Unfortunately, I didn’t find any other resource about this else where.
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It amazes me that derivations such as this are so hard to find. Some other information on this subject is here: https://vulcanhammernet.files.wordpress.com/2017/01/soilmechbook.pdf
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Thank you for sharing this with me. I will make sure to check it out.
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