“real: chapter_09” — 2011/5/22 — 23:21 — page 25 — #25
Sequences and Series of Functions 9-25
(a) {f
n
} is uniformly bounded on K.
(b) {f
n
} admits a subsequence that converges uniformly on K.
Proof The proof depends on the following steps:
Step 1: {f
n
} is uniformly bounded on K.
Step 2: There exists a countable dense subset E of K.
Step 3: {f
n
} admits a subsequence, say {f
n
k
= g
k
}, which converges
pointwise at every x ∈ E.
Step 4: {g
k
} converges uniformly on K.
Note that Step 1 and Step 4 are precisely the required conclusions.
We shall prove Step 1 independently and show that Step 2 and Step 3
together imply Step 4. Finally, we shall prove Step 2 and Step 3 to
complete the proof.
Proof of Step 1: Using equicontinuity of the family {f
n
}, given >0
we choose ...