Can "and" be Substituted with "+" in proofs

Question

Notes:

Considering two limit were given

$$\mathop {\lim }\limits_{x \to a} f\left( x \right) = L \hspace{0.1in} and \hspace{0.1in} \mathop {\lim }\limits_{x \to a} g\left( x \right) = M$$

means there is $\delta_1>0$ so that$ |f(x)-L|< \varepsilon$ whenever $<0|x-a|>\delta_1$

also there is $\delta_2>0 $ so that $|g(x)-M|< \varepsilon$ whenever $<0|x-a|>\delta_2$

choose $\delta=\min(\delta_1,\delta_2)$ then

$|f(x)-L|< \varepsilon$ and $|g(x)-M|< \varepsilon$ whenever $0<|x-a|<\delta$

Question:

Is it okay Substitute "and" with "+" so that

$|f(x)-L| + |g(x)-M|< \varepsilon + \varepsilon$ whenever $0<|x-a|<\delta$

Answer 1

Answer to the edited question:

If $|f(x)-L|<\varepsilon$ and $|g(x)-M|<\varepsilon,$ then it is valid to conclude that $|f(x)-L|+|g(x)-M|<\varepsilon+\varepsilon=2\varepsilon.$ This is an application of $a<b$ and $c<d \implies a+c<b+d$, which follows from $a+c<b+c$ and $b+c<b+d$.

Answer 2

No, do not do this. Write "and". Why would you need to save 2 letters anyway?

In fact, $$ |f(x)-L|< \varepsilon + |g(x)-M|< \varepsilon $$ already has a meaning. It represents two inequalities: $$ |f(x)-L|< \varepsilon + |g(x)-M|\qquad \text{and} \qquad \varepsilon + |g(x)-M| < \varepsilon $$

Answer 3

No, absolutely not. I get where you're coming from with the "$+$" as a substitute for "and" but it makes it look weirder, if not altogether wrong. For example if I had the two statements

$$1 < 2 \;\;\;\;\; \text{and} \;\;\;\;\; -3 < 1$$

but did this replacement here I would have

$$1 < 2 + -3 < 1$$

which looks like an intermediate step in saying $1<-1<1$ which is just plain nonsense.

An argument could be made for using an ampersand instead, "&", but I think something important to bear in mind is that in proofs you're usually expected to use words. Once you start doing proof-based things, you're also having to communicate ideas as well as the underlying mathematics, and for that you might want to write in prose as opposed to going down to shorthand. Not just to avoid ambiguities like the above but also just in the interest of communication.