# Squeeze (Sandwich) Theorem for Sequences

## Related Calculator: Limit Calculator

Consider two sequences x_n and y_n. When we write that x_n=y_n we mean that corresponding values are equal, i.e. x_1=y_1, x_2=y_2 etc.

Fact 1. If two sequences x_n and y_n are equal: x_n=y_n, and each of them has limit (finite or infinite): lim x_n=a and lim y_n=b then a=b.

This fact is used in limiting process: from x_n=y_n we conclude that lim x_n=lim y_n.

Fact 2. If for two sequences x_n and y_n we have that x_n>=y_n, and each of them has limit (finite or infinite): lim x_n=a and lim y_n=b then a>=b.

This fact is used in limiting process: from x_n>=y_n we conclude that lim x_n>=lim y_n.

Of course sign >= can be replaced by sign <=, i.e. from x_n<=y_n we conclude that lim x_n<=lim y_n.

CAUTION! Inequality x_n>y_n we can't conclude that lim x_n>lim y_n, we can only conclude that lim x_n>=lim y_n.

For example, consider two sequences x_n=1/n and y_n=-1/n. Clearly, x_n>y_n but lim 1/n=lim -1/n=0.

Fact 3 (Squeeze Theorem for Sequences). Consider three sequences x_n,y_n,z_n. If we have that x_n<=y_n<=z_n, and sequence x_n and z_n have same limit (finite or infinite), i.e. lim x_n=lim z_n=a, then lim y_n=a.

This theorem tells us folowing: if there are three sequences, two of which have same limit and third is "squeezed" between them, then third will have same limit as first two.

Consider figure to the right. Let sequence drawn in green and sequence drawn in blue have same limit a. Since sequence drawn in pink is "squeezed" between them, then it will also have limit a.

From this theorem it follows that if for all n a<=y_n<=z_n and z_n->a then y_n->a.