Is there something wrong with this proof?

**math9** · February 19th 2011, 03:20 PM

Someone please tell me if this proof is wrong:

QUESTION: Let $f_n:X\to\mathbb{R}$ be a sequence of functions such that $f_n(x)\neq 0$ for all $n\in\mathbb{N}$ and for all $x\in X$ . If there exists $c\in\mathbb{R}$ and $N\in\mathbb{N}$ such that $|\frac{f_{n+1}(x)}{f_n(x)}|\leq c<1$ for all $n\geq N$ and all $x\in X$ then $\sum |f_n(x)|$ and $\sum f_n(x)$ converge uniformly on X.

PROOF: Let $x\in X$ and $n\geq N$ . Then $|\frac{f_{N+1}(x)}{f_N(x)}|\leq c$ , $|\frac{f_{N+2}(x)}{f_{N+1}(x)}|\leq c$ , ... , $|\frac{f_{n}(x)}{f_{n-1}(x)}|\leq c$ . Multiplying all these inequalities we get $|\frac{f_{n}(x)}{f_N(x)}|\leq c^{n-N}\Rightarrow |f_n(x)|\leq c^{n-N}|f_N(x)|$ . If we put $a_n=c^{n-N}|f_N(x)|$ then the conclusion follows from the Weierstrass M-Test, since $\sum a_n$ is convergent.

**Opalg** · February 20th 2011, 12:11 AM

Originally Posted by math9

Someone please tell me if this proof is wrong:

QUESTION: Let $f_n:X\to\mathbb{R}$ be a sequence of functions such that $f_n(x)\neq 0$ for all $n\in\mathbb{N}$ and for all $x\in X$ . If there exists $c\in\mathbb{R}$ and $N\in\mathbb{N}$ such that $|\frac{f_{n+1}(x)}{f_n(x)}|\leq c<1$ for all $n\geq N$ and all $x\in X$ then $\sum |f_n(x)|$ and $\sum f_n(x)$ converge uniformly on X.

PROOF: Let $x\in X$ and $n\geq N$ . Then $|\frac{f_{N+1}(x)}{f_N(x)}|\leq c$ , $|\frac{f_{N+2}(x)}{f_{N+1}(x)}|\leq c$ , ... , $|\frac{f_{n}(x)}{f_{n-1}(x)}|\leq c$ . Multiplying all these inequalities we get $|\frac{f_{n}(x)}{f_N(x)}|\leq c^{n-N}\Rightarrow |f_n(x)|\leq c^{n-N}|f_N(x)|$ . If we put $a_n=c^{n-N}|f_N(x)|$ then the conclusion follows from the Weierstrass M-Test, since $\sum a_n$ is convergent.

In the last line of the proof, you can't apply the M-test in that way, because $a_n$ depends on $x$ . The M-test only applies if $(a_n)$ is a sequence of constants. If the function $f_N(x)$ is bounded, say $|f_N(x)|\leqslant K$ for all $x\in X$ , then you can take $a_n = c^{n-N}K$ . In that case, the M-test does apply and the conclusion is correct. In particular, if the space $X$ is compact, then $f_N(x)$ will automatically be bounded and the result is correct.

But suppose for example that $X=\mathbb{R}$ and $f_n(x) = 2^{-n}x$ . Then $\sum_{n=1}^\infty f_n(x) = x$ for all $x\in\mathbb{R}$ , but the convergence is not uniform.

**math9** · February 20th 2011, 04:24 AM

Originally Posted by Opalg

In the last line of the proof, you can't apply the M-test in that way, because $a_n$ depends on $x$ . The M-test only applies if $(a_n)$ is a sequence of constants. If the function $f_N(x)$ is bounded, say $|f_N(x)|\leqslant K$ for all $x\in X$ , then you can take $a_n = c^{n-N}K$ . In that case, the M-test does apply and the conclusion is correct. In particular, if the space $X$ is compact, then $f_N(x)$ will automatically be bounded and the result is correct.

But suppose for example that $X=\mathbb{R}$ and $f_n(x) = 2^{-n}x$ . Then $\sum_{n=1}^\infty f_n(x) = x$ for all $x\in\mathbb{R}$ , but the convergence is not uniform.

I see.... Do you have any suggestions as to how I can fix my proof? Or maybe another proof?

**Opalg** · February 20th 2011, 05:11 AM

Originally Posted by math9

I see.... Do you have any suggestions as to how I can fix my proof? Or maybe another proof?

No. Unless you have some extra condition to ensure that $f_N(x)$ is bounded, the result is false (as the example at the end of my previous comment shows).

**math9** · February 20th 2011, 07:51 AM

Originally Posted by Opalg

No. Unless you have some extra condition to ensure that $f_N(x)$ is bounded, the result is false (as the example at the end of my previous comment shows).

Well.. the example you gave doesn't really show the result is false since your sequence of functions take on the value 0 at x=0 and one of the hypothesis of the problem is that $f_n(x)\neq0$ for all $x \in X$ .

**Opalg** · February 20th 2011, 08:18 AM

Originally Posted by math9

Well.. the example you gave doesn't really show the result is false since your sequence of functions take on the value 0 at x=0 and one of the hypothesis of the problem is that $f_n(x)\neq0$ for all $x \in X$ .

I can easily fix that by restricting the domain to the interval $X = [1,\infty)$ . The hypotheses of the question are then all satisfied, but $\sum f_n(x)$ does not converge uniformly on $X$ .

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