Proof of 1 = 0 by Mathematical Induction on Limits? I got stuck with a problem that pop up in my mind while learning limits. I am still a high school student. 
Define $P(m)$ to be the statement: $\quad \lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{m})=0$
The statement holds for $m = 1$: $\quad \lim\limits_{n\to\infty}\frac{1}{n}=0$.
Assume that $P(k)$ holds for some $k$. So put $m = k$: $\quad \lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{k})=0$.
We prove $P(k + 1)$: $\quad \lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{k+1}) =\lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{k}+\frac{1}{n})$
$=\lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{k}) +\lim\limits_{n\to\infty}\frac{1}{n}$
$=0+0=0$.
It has now been proved by mathematical induction that statement holds for all natural m.
If we let $m=n$, then $\lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{n})=0 \tag{*}$.
However, $\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{n}=1 \implies \lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{n})=1 \tag{$\dagger$}$.
Then $(*) \, \& \, (\dagger)$ yield $1=0$?
Can anybody explain this? thanks.
 A: The problem is that the statement you proved was for fixed $m$, and then you let it vary.
What follows is one way of looking at this problem:
Rewrite things as: $$\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{m}=\frac{m}{n}$$
Then what you proved by induction is that for any fixed $m$ $$\lim_{n\rightarrow \infty}\frac{m}{n}=\left(\lim_{n\rightarrow \infty}m\right)\cdot\left(\lim_{n\rightarrow \infty}\frac{1}{n}\right)=m\cdot 0=0.$$  This is fine since when the limits exist we can split them up like above.  However in the second deduction you try to do the same thing
$$\lim_{n\rightarrow \infty}\frac{n}{n}=\left(\lim_{n\rightarrow \infty}n\right)\cdot\left(\lim_{n\rightarrow \infty}\frac{1}{n}\right)=n\cdot 0=0.$$ This doesn't make any sense now, because $n$ is no longer fixed, and the one limit does not exist.  (We only have the multiplicative property when both limits exist)
Hope that helps,
A: $n$ is a free variable of the term $n$ that becomes bound during the substitution $m=n$ into
$\lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{m})=0$
so the substitution is not logically valid.
