- #1
phydis
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Homework Statement
a<b<c and, f is bounded on [a,b] and f is bounded on [b,c] prove that f is bounded on [a,c]
The Attempt at a Solution
there exist M1≥0 s.t. for all x ε [a,b] |f(x)|≤M1
there exist M2≥0 s.t. for all x ε [b,c] |f(x)|≤M2
for x ε [a,b] and x ε [b,c]
Let M>0, and let M>M1 and M>M2
therefore
|f(x)|≤M1<M --> |f(x)|<M and |f(x)|≤M2<M --> |f(x)|<M
∴ there exist M>0 s.t. |f(x)|<M *
so f is bounded on [a,c]
is this proof correct? definition says f is bounded on [a,c] if M≥0 s.t. for all x ε [a,c] |f(x)|≤M
but what I have proven is, f is bounded on [a,c] since M>0 s.t. for all x ε [a,c] |f(x)|<M