Question

Can anyone prove the following?
Suppose each of S and T is a bounded set. Prove S union T is bounded

Answer 1

What does bounded mean?

Answer 2

"If s is a set of number then the statement that S is bounded about means there exists a number that if x is in S, then x <= M. Such a number M is called an upper bound for the set S"
That's all i got...

Answer 3

Oh well. Since S is bounded, then there exist a and b such that for all s in S
s ≤ b (its upper bound) and
s ≥ a (its lower bound)

Similarly, since T is bounded, then there exist c and d such that for all t in T
t ≤ d (its upper bound) and
t ≥ c (its lower bound)

define v=max(b, d), [in other words, whichever is greater between b and d]
define u=min(a, c), [in other words, whichever is lesser between a and c]

Then
b ≤ v AND d ≤ v
Also
a ≥ u AND c ≥ u

Now, let k be an element of S ∪ T
Then either k is in S OR k is in T.
Suppose k is in S.
Then
k ≥ a ≥ u, therefore, k ≥ u.
and
k ≤ b ≤ v, therefore, k ≤ v.
then
u ≤ k ≤ v

Suppose k is in T.
Then
k ≥ c ≥ u, therefore, k ≥ u.
and
k ≤ d ≤ v, therefore, k ≤ v.
then
u ≤ k ≤ v

In both cases,
u ≤ k ≤ v

Therefore,
S ∪ T is bounded below by u and bounded above by v.
Hence
S ∪ T is bounded
∎

Answer 4

wow. Thanks!

Answer 5

No sweat :)