always begin with the definition of the relevant term. e.g. to prove that a certain collecion of vectors is linearly independent, recall first the definition of liner independence.
e.g. to prove that the sequence (1,0,0), (1,1,0), 1,1,1) is linearly independent, (over the rationals say), recall that a sequence v1,v2,v3 is linearly independent over Q if and only if the only rational scalars a1,a2,a3 such that a1v1 + a2v2 + a3v3 = 0 (the zero vector), are the scalars a1 = a2 = a3 = 0.
Then phrase your problem in these terms: i.e. hence to prove that our sequence (1,0,0), (1,1,0), 1,1,1) is linearly independent over Q we must show that the only rational numbers a1,a2,a3 such that a1(1,0,0) + a2(1,1,0) + a3(1,1,1) = (0,0,0), are the rationals a1 = a2 = a3 = 0.
Now go to work calculating with your example: i.e. we are assuming that a1(1,0,0) + a2(1,1,0) + a3(1,1,1) = (0,0,0), and we want to deduce that a1 = a2 = a3 = 0.
If we calculate we get starting from our assumption, that (0,0,0) = a1(1,0,0) + a2(1,1,0) + a3(1,1,1) = (now calculate) (a1,0,0) + (a2,a2,0) + (a3,a3,a3)
= (a1+a2+a3, a2+a3, a3). Now look at your calculation and try to see how to deduce your result:
I.e. since we have (0,0,0) = (a1+a2+a3, a2+a3, a3), looking at the last entry we see that a3 = 0. Substituting that value gives (0,0,0) = (a1+a2, a2, 0). Now looking at the second entries gives a2 = 0. Substituting again gives (0,0,0) = (a1,0,0), hence a1 = 0 also. Hence we have a1 = a2 = a3 = 0.I have written this proof for the absolute beginner, who needs every step spelled out from definition to conclusion. If you are writing for more experienced readers, you tend to skip more and mopre of these steps, but this is the way to begin.
But to re-iterate: a proof is a logical argument leading from a definition to a specific conclusion, hence it is always prudent to begin by stating, then specializing, the relevant definition involved. I.e. you cannot prove the truth of a statement involving technical words unless you know exactly what those words mean.