- #1

DaTario

- 1,039

- 35

- Homework Statement
- Let ##\sum_{i=1}^N a_{i,N} /N = 1## with ## 0 < a_{i,N} < M > 1 ## for all ##i## and ##N##.

Show that ##\lim_{N \to \infty} \sum_{i=1}^N (a_i /N)^2 = 0##.

- Relevant Equations
- We know that each ##a_{i,N}/N## is positive and less than one implying that their square is even smaller.

I don't know how to show that this limit is zero.

It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one.

Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.

It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one.

Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.