Basic analysis: Proof by induction on sets

[K29]

Homework Statement

Prove by induction that the set [a_{n} | n_{0}\leq n \leq n_{1}] is bounded.
a_{n} are the elements of the sequence (a_{n})
n \in N

Homework Equations

Definition of set bounded above:
\forall x \in S, \exists M \in R such that x \leq M

The Attempt at a Solution

Just proving its bounded above here...

Base step: [a_{1}] The set has only 1 element and
a_{1} \leq a_{1} +1

Now assume true for a_{n}
[a_{n}|n_{0}\leq n \leq n_{1}]
and \exists M \in R such that a_{n} \leq M, <br /> forall a_{n}

For [a_{n+1}|n_{0} \leq n+1 \leq n_{1} ]
... well I'm not really sure what to do here. Normally you use the assumption to prove it true for n+1, but I'm not sure how to incorporate the assumption here.
Please help

 

[Stephen Tashi]

Now assume true for a_{n}
[a_{n}|n_{0}\leq n \leq n_{1}]
and \exists M \in R such that a_{n} \leq M,
for all a_{n}

For [a_{n+1}|n_{0} \leq n+1 \leq n_{1} ]
... well I'm not really sure what to do here

To do induction, you should consider the set \{ a_n | n_0 \leq n \leq n_1+1 }
Say its bound is max{a_{n+1}, M}.

"Bounded" might mean bounded below and above. If it means that in the problem, you should also do the workd for "bounded below".

 

[K29]

Great stuff. solved. Thanks!