# How I can sketch the proof of consistency of only one beta in multiple regression?

Now assume you additionally obtained data on average parental incomes (PI) and the ethnic composition (EC) of the pupils in school. You regress the score on

• STR
• PI
• EC and
• a constant.

State the main assumption for consistent estimation of the coefficient on STR. Assume a linear setting and sketch a proof for consistency of the OLS estimator for the coefficient on STR.

Normally, I use matrix form for this kind of question, then how can I prove for only one beta, I was thinking about Frisch-Waugh-Theorem but i cannot move on to proof it. Any idea ?