information:extraction
Pointwise limits and uniform limits

For a double sequence (x n, m) {\displaystyle (x_{n,m})}, we may take limit in one of the indices, say, n → ∞ {\displaystyle n\to \infty }, to obtain a single sequence (y m) {\displaystyle (y_{m})}. In fact, there are two possible meanings when taking this limit. The first one is called pointwise limit, denoted

which means:

Symbolically, this is:

When such a limit exists, we say the sequence (x n, m) {\displaystyle (x_{n,m})} converges pointwise to (y m) {\displaystyle (y_{m})}.

The second one is called uniform limit, denoted

which means:

Symbolically, this is:

In this definition, the choice of N {\displaystyle N} is independent of m {\displaystyle m}. In other words, the choice of N {\displaystyle N} is uniformly applicable to all natural numbers m {\displaystyle m}. Hence, one can easily see that uniform convergence is a stronger property than pointwise convergence: the existence of uniform limit implies the existence and equality of pointwise limit:

When such a limit exists, we say the sequence (x n, m) {\displaystyle (x_{n,m})} converges uniformly to (y m) {\displaystyle (y_{m})}.