Download How To Prove That Two Things Are Equal

How To Prove That Two Things Are Equal
When trying to prove that two quantities are equal, the best way to proceed is to work on
scratch paper first. Manipulate the two things separately to see if you can find matching
renditions of the two quantities. Then, when you want to write up your proof, start with
one of the quantities and manipulate it until it becomes the other.
It is not advisable to write up your proof in a format that manipulates both quantities
simultaneously while joining the two quantities by the equals symbol.
Example from §13.3/14.3, problem #35c: Prove that
~
~
~
lim (~u(t) · v(t)) = lim u(t) · lim v(t) .
t→a
t→a
t→a
Here’s how I would write this up.
The idea of the argument is to write everything out in components. For ease of display,
let’s suppose that ~u and ~v are functions with vector values in R2 , say ~u(t) = hu1 (t), u2 (t)i
and ~v(t) = hv1 (t), v2 (t)i. We start with the left-hand-side of the equation to be verified and
proceed one step at a time.
lim (~u(t) · ~v(t)) = limhu1 (t), u2 (t)i · hv1 (t), v2 (t)i
t→a
(1)
t→a
= lim (u1 (t)v1 (t) + u2 (t)v2 (t))
t→a
= lim u1 (t)v1 (t) + lim u2 (t)v2 (t)
t→a
t→a
= lim u1 (t) lim v1 (t) + lim u2 (t) lim v2 (t)
t→a
t→a
t→a
t→a
= hlim u1 (t), lim u2 (t)i · hlim v1 (t), lim v2 (t)i
t→a
t→a
t→a
t→a
= lim ~u(t) · lim ~v(t)
t→a
t→a
(2)
(3)
(4)
(5)
(6)
Thus, the limit of the dot product of two vector-valued functions is the dot product of the
limits.
Comments: Notice how the derivation begins with one of the quantities (the left side in this
case) and then proceeds step-by-step using known equalities until the right hand side appears.
For example, line (1) simply expands the vector functions ~u and ~v in components. From (1)
to (2) we invoke the definition of the dot product. From (2) to (3) we use the known fact
that for real-valued functions, the limit of a sum is the sum of the limits. From (3) to (4)
we use that the limit of a product is the product of the limits. From (4) to (5) we use the
definition of the dot product again. From (5) to (6) we use the definition of the limit of a
vector-valued function. When you write up solutions, it is not necessary, though it may be
helpful, to annotate numerically in this fashion. However, you should have the justifications
for each step clearly in your mind. When a step is tricky, then it’s good practice to provide
some explanation for your reader.
If you choose to carry out the argument using vector-valued functions in three (or more)
components, the displayed sums may become unwieldy, though quite repetitive. This suggests that it may be convenient to use summation notation. For example,
lim u1 (t) lim v1 (t) + lim u2 (t) lim v2 (t) + lim u3 (t) lim v3 (t)
t→a
t→a
t→a
t→a
t→a
t→a
can be written much more compactly as
3 X
i=1
lim ui (t) lim vi (t) .
t→a
t→a
I want to contrast the above argument with one that appears frequently, but can be
misleading and sometimes erroneous. In this approach, common in student papers, the
author writes down the equation to be verified and then works simultaneously with both
sides, all the while displaying the equations as if they were true.
lim (~u(t) · ~v(t)) =
t→a
lim ~u(t) · lim ~v(t)
t→a
t→a
limhu1 (t), u2 (t)i · hv1 (t), v2 (t)i = hlim u1 (t), lim u2 (t)i · hlim v1 (t), lim v2 (t)i
t→a
t→a
t→a
t→a
t→a
lim(u1 (t)v1 (t) + u2 (t)v2 (t)) = lim u1 (t) lim v1 (t) + lim u2 (t) lim v2 (t)
t→a
t→a
t→a
t→a
t→a
lim u1 (t) lim v1 (t) + lim u2 (t) lim v2 (t) = lim u1 (t) lim v1 (t) + lim u2 (t) lim v2 (t)
t→a
t→a
t→a
t→a
t→a
t→a
t→a
t→a
When people do things like this, the logic implied is that since the separate manipulations
produced the same thing on both sides at the end, the things on both sides at the beginning
must be equal as well. There are two reasons to be concerned about this style of argument
and display. First, it is difficult to ascertain what path the author is taking. The author is
proceeding down the left side from step to step, but there is nothing on the page to indicate
this. Thus the train of thought is unclear to the reader. Second, in order for the logic to be
sound, we must actually determine whether each of the steps taken is reversible! That is, we
are trying to conclude that the beginning things were equal from the fact that the ending
things are identical. The danger here is that not every manipulation or logical implication is
reversible. As a silly example, here’s how to ”prove” that 1 = -1 using this sort of backwards
argument.
1 = −1
(1)2
=
(−1)2
1
=
1
So the backwards style of argument would encourage us to conclude that since the two things
at the end are equal, so must the things at the beginning be equal. Of course this is incorrect
and the problem is that while squaring equal numbers yields equal results, it is simply not
true that two numbers with the same square must be equal. So when you are trying to
prove the truth of an equation, don’t start out by writing the equation down as if you are
assuming that it is true. You’ll be confusing your reader and possibly setting yourself up for
an embarrassingly circular argument.