What is the importance of the word "inductively"? can I omit it?

Question

Lemma 2.2.2.: For any natural number $n$, $n + 0 = n$.
Note that we cannot deduce this immediately from $0+m = m$ because we do not know yet that $a + b = b + a.$
Proof: We use induction. The base case $0 + 0 = 0$ follows since we know that $0 + m = m$ for every natural number $m$, and $0$ is a natural number. Now suppose inductively that $n + 0 = n$. We wish to show that $(n++) + 0 = n++$. But by definition of addition, $( n++) + 0$ is equal to $( n + 0) ++$, which is equal to $n++$ since $n + 0 = n$. This closes the induction.
what is the meaning of the word "inductively" when it says "suppose inductively"?

Answer 1

The meaning of the phrase "inductively" in this context is something like this.

The induction step of any induction proof is stated in the following manner:

If $P(n)$ is true then $P(n+1)$ is true.

The first portion of this, namely the phrase "$P(n)$ is true", is called the induction hypothesis.

In your proof, here's what you actually write to start the proof of the induction step:

Assume that $P(n)$ is true.

And then you would continue with

BLAH BLAH BLAH (lots of details of proof) BLAH BLAH BLAH therefore $P(n+1)$ is true.

Of course the flexibility of the English language let's you write things in other ways, and that first sentence of that proof can be written in other ways. For instance you might have a more verbose tendency and write

Assume that the inductive hypothesis $P(n)$ is true.

Or, you might have a less verbose tendency and write

Assume inductively $P(n)$.

That's all that's going on, its just another way to actually start the proof of the induction step.

Answer 2

In my opinion it is slightly strange to say "suppose inductively" and refer to induction when making the hypothesis, and it is more common to say "and then inductively a result follows" and refer to induction while making the step argument.

In this case they are assuming $n+0=n$ as the hypothesis for the induction step. We assume it with a strategy in mind-- we are going to show $(n++)+0=n++$-- so I think the author means "we are assuming it for inductive purposes".

We are certainly not assuming it in any inductive manner which is what an adverb such as "inductively" would imply.

I personally would use "inductively" when we make an argument. For examply I might say: "If $b>1$ then $b*b > b*1$ and $b*1>b$ and inductively we can conclude $1 < b < b^2 < .....<b^k < b^{k+1}< ......$"