- #1
evagelos
- 315
- 0
formal vs informal
in proving that: [itex]lim_{n\rightarrow\infty} \frac{1}{n}\neq 1[/itex] the following proof was suggested.
Proof:
Suppose [itex]lim_{n\rightarrow\infty}\frac{1}{n} =1[/itex], but [itex]lim_{n\rightarrow\infty}\frac{1}{n} =0[/itex], hence:
For all ε>0
1) There exists mεN such that: [itex]n\geq m\Longrightarrow |\frac{1}{n}|<\frac{\epsilon}{2}[/itex]
2)There exists kεN such that : [itex]n\geq k\Longrightarrow |\frac{1}{n}-1|<\frac{\epsilon}{2}[/itex]
Choose r = max{m,k},then [itex]r\geq m,r\geq k[/itex]
Let ,[itex]n\geq r\Longrightarrow n\geq m\wedge n\geq k[/itex].
Hence : [itex]|\frac{1}{n}|<\frac{\epsilon}{2}[/itex] and [itex]|\frac{1}{n}-1|<\frac{\epsilon}{2}[/itex].
Thus : [itex]|\frac{1}{n}-\frac{1}{n}+1|=1\leq |\frac{1}{n}| + |\frac{1}{n}-1|<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon[/itex]
.....or 1<ε...But since this holds for all ε>0 we put ε=1 and we have 1<1 ,a contradiction .
Therefor [itex]lim_{n\rightarrow\infty}\frac{1}{n}\neq 1[/itex]
Write a formal proof of the above ,thus proving that the above informal proof is wrong
in proving that: [itex]lim_{n\rightarrow\infty} \frac{1}{n}\neq 1[/itex] the following proof was suggested.
Proof:
Suppose [itex]lim_{n\rightarrow\infty}\frac{1}{n} =1[/itex], but [itex]lim_{n\rightarrow\infty}\frac{1}{n} =0[/itex], hence:
For all ε>0
1) There exists mεN such that: [itex]n\geq m\Longrightarrow |\frac{1}{n}|<\frac{\epsilon}{2}[/itex]
2)There exists kεN such that : [itex]n\geq k\Longrightarrow |\frac{1}{n}-1|<\frac{\epsilon}{2}[/itex]
Choose r = max{m,k},then [itex]r\geq m,r\geq k[/itex]
Let ,[itex]n\geq r\Longrightarrow n\geq m\wedge n\geq k[/itex].
Hence : [itex]|\frac{1}{n}|<\frac{\epsilon}{2}[/itex] and [itex]|\frac{1}{n}-1|<\frac{\epsilon}{2}[/itex].
Thus : [itex]|\frac{1}{n}-\frac{1}{n}+1|=1\leq |\frac{1}{n}| + |\frac{1}{n}-1|<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon[/itex]
.....or 1<ε...But since this holds for all ε>0 we put ε=1 and we have 1<1 ,a contradiction .
Therefor [itex]lim_{n\rightarrow\infty}\frac{1}{n}\neq 1[/itex]
Write a formal proof of the above ,thus proving that the above informal proof is wrong
Last edited: