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## Homework Statement

In usual textbooks, what are the meanings of the notations

(1) [itex]\sup_{k\in\mathbb{N}}|x_{k}|[/itex]

(2) [itex]\sup_{x\in [ a , b ] }|f(x)-g(x)|[/itex]

where [itex](x_{k})[/itex] is a real sequence and [itex]f[/itex] and [itex]g[/itex] are real valued functions

## Homework Equations

None.

## The Attempt at a Solution

The meaning I guessed is that

(1) [itex]\sup_{k\in\mathbb{N}}|x_{k}|=\sup\{|x_{k}|:k\in \mathbb{N}\}[/itex]

(2) [itex]\sup_{x\in[a,b]}|f(x)-g(x)|=\sup\{|f(x)-g(x)|:x\in[a,b]\}[/itex].