Hello I'm reading Lang's Intro to Linear Algebra & I've noticed that he uses squaring an equation,

working out the algebra and then square-rooting to prove a theorem.

I'm trying to get used to proofing for analysis &

I'd like to know whether squaring is considered an adequate method of proof?

A quick example is;

||xA|| = |x| ||A||

**1/** ||xA||² = {√[(xA) · (xA)]}² = xA · xA

* (Using the definition ||A|| = √(A · A) = √(a_1² + a_2² + ... + a_n²)* **2/** xA · xA = (xa_1, xa_2, ..., xa_n) · (xa_1, xa_2, ..., xa_n)

* (Using the definition of vector A : (a_1, a_2, ..., a_n) )* **3/** x²a_1² + x²a_2² + ... + x²a_n² = x²(a_1² + a_2² + ... + a_n²)

*(Using the Dot Product property for components)* **4/** x² A · A = |x|² ||A||²

*(Rewriting the squared components in ***3/** as *A · A,*

*x² as |x|² to account for ± x values & A · A as ||A||² *

as just representing the definition of ||A|| squared).

**5/ **Take a square root, and voila!

Here he set out to prove his theorem by squaring out one side,

working out the algebra and achieving the other side i.e. proving an equality.

I'm just concerned as to whether this would constitute a rigid proof

in an analysis book as the ones I've read

*(albeit I was lacking the mathematical maturity I have now, *__which is still in it's infancy!__)
would seemingly come out of nowhere