Suppose-I-want-to-prove-a-statement-say-P-n-for-natural-numbers-mathematical-induction-is-a-proper-tool-for-me-If-a-P-n-is-required-to-prove-for-natural-n-c-mathematical-induction-is-again-a-too

Question Number 2936 by Rasheed Soomro last updated on 30/Nov/15

S u p p o s e I w a n t t o p r o v e a s t a t e m e n t, s a y P (n),

f o r n a t u r a l n u m b e r s m a t h e m a t i c a l i n d u c t i o n

i s a p r o p e r t o o l f o r m e .

I f a P (n) i s r e q u i r e d t o p r o v e f o r n a t u r a l n ⩾ c

m a t h e m a t i c a l i n d u c t i o n i s a g a i n a t o o l o f p r o o f .

N o w s u p p o s e I h a v e a s t a t e m e n t w h i c h i s t r u e

o n l y f o r \underline{c_{1} ⩽ n \in N ⩽ c_{2}} . C o u l d I g e t h e l p f o r p r o o f

f r o m m y o l d f r i e n d m a t h e m a t i c a l i n d u c t i o n ?

I f y e s h o w ?

Commented by Filup last updated on 01/Dec/15

I belive mathematical induction is used

to prove \forall n \in P (n) : {n \in N ⩾ c_{1}} .

I am unsure as to if you can prove within

the bounds such that you can show

\forall n \in P (n) : {c_{1} ⩽ n \in N ⩽ c_{2}} .

If it was possible, it may be wise to work

with and solve for :

i) \forall n \notin P (n) : {n \in N ⩽ c_{1}}

i i) \forall n \notin P (n) : {n \in N ⩾ c_{2}}

Commented by Filup last updated on 01/Dec/15

I {don}^{'} t know much of this topic to be

of any help . S o r r y

Commented by Rasheed Soomro last updated on 01/Dec/15

Y e s F i l u p y o u a r e r i g h t .

I n t h e m e n t i o n e d c a s e m a t h e m a t i c a l i n d u c t i o n

{c o u l d n}^{'} t h e l p .

Commented by 123456 last updated on 02/Dec/15

we can use it to proof things to Z

P (n) \to P (n + 1)

P (n) \to P (n - 1)

i think that there some way to use it

c_{1} ⩽ n ⩽ c_{2}

but probraly the bounds will be involved

into inductive step

Commented by Rasheed Soomro last updated on 02/Dec/15

C o u l d y o u c l e a r w i t h a n e x a m p l e ?